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There exist a variety of formulas for either producing the th prime as a function of or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers ), there exists a constant (OEIS A051021) known as Mills' constant such that(1)where is the floor function, is prime for all (Ribenboim 1996, p. 186). The first few values of are 2, 11, 1361, 2521008887, ... (OEIS A051254). It..

The prime distance of a nonnegative integer is the absolute difference between and the nearest prime. It is therefore true that for primes . The first few values for , 1, 2, ... are therefore 2, 1, 0, 0, 1, 0, 1, 0, 1, 2, ... (OEIS A051699). The values of having prime distances of 0, 1, 2, 3, ... are 2, 1, 0, 26, 93, 118, 119, 120, 531, 532, 897, ... (OEIS A077019).

If and (i.e., and are relatively prime), then has at least one primitive prime factor with the following two possible exceptions: 1. . 2. and is a power of 2. Similarly, if , then has at least one primitive prime factor with the exception .A specific case of the theorem considers the th Mersenne number , then each of , , , ... has a prime factor that does not occur as a factor of an earlier member of the sequence, except for . For example, , , , ... have the factors 3, 7, 5, 31, (1), 127, 17, 73, 11, , ... (OEIS A064078) that do not occur in earlier . These factors are sometimes called the Zsigmondy numbers .Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same (Montgomery 2001)...

A prime factorization algorithm also known as Pollard Monte Carlo factorization method. There are two aspects to the Pollard factorization method. The first is the idea of iterating a formula until it falls into a cycle. Let , where is the number to be factored and and are its unknown prime factors. Iterating the formula(1)or almost any polynomial formula (an exception being ) for any initial value will produce a sequence of number that eventually fall into a cycle. The expected time until the s become cyclic and the expected length of the cycle are both proportional to .However, since with and relatively prime, the Chinese remainder theorem guarantees that each value of (mod ) corresponds uniquely to the pair of values (), ). Furthermore, the sequence of s follows exactly the same formula modulo and , i.e.,(2)(3)Therefore, the sequence (mod ) will fall into a much shorter cycle of length on the order of . It can be directly verified that two values and..

A variant of the Pollard p-1 method which uses Lucas sequences to achieve rapid factorization if some factor of has a decomposition of in small prime factors.

Bertelsen's number is an erroneous name erroneously given to the erroneous value of , where is the prime counting function. This value is 56 lower than the correct value of . Ore (1988, p. 69) states that the erroneous value 478 originated in Bertelsen's application of Meissel's method in 1893 (MathPages; Prime Curios!). However, the incorrect value actually first appears in Meissel (1885) rather than Bertelsen in 1893, as correctly noted by Lagarias et al. 1985. (Note that MathPages incorrectly states that Lagarias et al. attribute the result to Bertelsen.)Unfortunately, the incorrect value has continued to be propagated in modern works such as Hardy and Wright (1979, p. 9), Davis and Hersch (1981, p. 175; but actually given correctly in the table on p. 213), Sondheimer (1981), Kramer (1983), Ore (1988, p. 77), and Cormen et al. (1990)...

(1)The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEISA001223). Rankin has shown that(2)for infinitely many and for some constant (Guy 1994). At a March 2003 meeting on elementary and analytic number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that(3)(Montgomery 2003). Unfortunately, this proof turned out to be flawed.An integer is called a jumping champion if is the most frequently occurring difference between consecutive primes for some (Odlyzko et al.).

Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

Let denote the th prime, and writeThen the exponent vector is .

Also known as the difference of squares method. It was first used by Fermat and improved by Gauss. Gauss looked for integers and satisfyingfor various moduli . This allowed the exclusion of many potential factors. This method works best when factors are of approximately the same size, so it is sometimes better to attempt for some suitably chosen value of .

An ordered factorization is a factorization (not necessarily into prime factors) in which is considered distinct from . The following table lists the ordered factorizations for the integers 1 through 10.#ordered factorizations11121231342, 451563, , 671784, , , 892, 9103, , 10The numbers of ordered factorizations of , 2, ... are given by 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS A074206). This sequence has the Dirichlet generating function(1)where is the Riemann zeta function.A recurrence equation for is given by(2)where the sum is over the divisors of and (Hille 1936, Knopfmacher and Mays 2006). Another recurrence also due to Hille (1936) for is given by(3)where and(4)is the prime factorization of (Knopfmacher and Mays 1996).MacMahon (1893) derived the beautiful double sum formula(5)where(6)(Knopfmacher and Mays 1996). In the case that is a product of two prime powers,(7)Chor et al. (2000) showed that(8)(9)where is a hypergeometric function.The..

A factorization algorithm which works by expressing as a quadratic form in two different ways. Then(1)so(2)(3)Let be the greatest common divisor of and so(4)(5)(6)(where denotes the greatest common divisor of and ), and(7)But since , and(8)which gives(9)so we have(10)(11)(12)(13)(14)(15)

An unordered factorization is a factorization of a number into a product of factors where order is ignored. The following table lists the unordered factorizations of the first few positive integers.unordered factorizations1122334, 4556, 6778, , 89, 910, 10A recurrence product for the number of unordered factorizations is given by Harris and Subbarao (1991).The numbers of unordered factorizations for , 2, ... are therefore 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, ... (OEIS A001055). The maximum numbers of parts in the unordered (or ordered) factorizations of for , 2, ... are 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, ... (OEIS A086436).The following gives a table of unordered factorizations with distinct parts for between 1 and 10.distinct unordered factorizations11223344556, 6778, 89910, 10The numbers of unordered factorizations with distinct parts for , 2, ... are given by 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, ... (OEIS A045778). The..

A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if is a prime and , then or (where means divides). A corollary is that (Conway and Guy 1996). The fundamental theorem of arithmetic is another corollary (Hardy and Wright 1979).Euclid's second theorem states that the number of primes is infinite. This theorem, also called the infinitude of primes theorem, was proved by Euclid in Proposition IX.20 of the Elements (Tietze 1965, pp. 7-9). Ribenboim (1989) gives nine (and a half) proofs of this theorem. Euclid's elegant proof proceeds as follows. Given a finite sequence of consecutive primes 2, 3, 5, ..., , the number(1)known as the th Euclid number when is the th prime, is either a new prime or the product of primes. If is a prime, then it must be greater than the previous primes, since one plus the product of primes must be greater than each prime composing the product. Now, if is a product of primes,..

In order to find integers and such that(1)(a modified form of Fermat's factorization method), in which case there is a 50% chance that is a factor of , choose a random integer , compute(2)and try to factor . If is not easily factorable (up to some small trial divisor ), try another . In practice, the trial s are usually taken to be , with , 2, ..., which allows the quadratic sieve factorization method to be used. Continue finding and factoring s until are found, where is the prime counting function. Now for each , write(3)and form the exponent vector(4)Now, if are even for any , then is a square number and we have found a solution to (◇). If not, look for a linear combination such that the elements are all even, i.e.,(5)(6)Since this must be solved only mod 2, the problem can be simplified by replacing the s with(7)Gaussian elimination can then be used tosolve(8)for , where is a vector equal to (mod 2). Once is known, then we have(9)where the products are taken..

Squarefree factorization is a first step in many factoring algorithms. It factors nonsquarefree polynomials in terms of squarefree factors that are relatively prime. It can separate factors of different multiplicities, but not factors with the same multiplicity. One way to find a squarefree factorization is to compute polynomial greatest common denominators iteratively.The squarefree part (i.e., product of all distinct monic irreducible factors) of a monic nonconstant polynomial in a field of characteristic zero is , where is the derivative of .

Let be any integer and let (also denoted ) be the least integer greater than 1 that divides , i.e., the number in the factorizationwith for . The least prime factor is implemented in the Wolfram Language as FactorInteger[n][[1,1]].For , 3, ..., the first few are 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, ... (OEIS A020639).If is composite then (Séroul 2000, p. 7), with equality for the square of a prime.A plot of the least prime factor function resembles a jagged terrain of mountains, which leads to the appellation of "twin peaks" to a pair of integers such that 1. , 2. , 3. For all , implies . The least multiple prime factors for squarefulintegers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, ... (OEIS A046027).Erdős et al. (1993) consider the least prime factor of the binomial coefficients, and define what they term good binomial coefficients and exceptional binomial coefficients. They also conjecture that..

The distinct prime factors of a positive integer are defined as the numbers , ..., in the prime factorization(1)(Hardy and Wright 1979, p. 354).A list of distinct prime factors of a number can be computed in the Wolfram Language using FactorInteger[n][[All, 1]], and the number of distinct prime factors is implemented as PrimeNu[n].The first few values of for , 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (OEIS A001221; Abramowitz and Stegun 1972, Kac 1959). This sequence is given by the inverse Möbius transform of , where is the characteristic function of the prime numbers (Sloane and Plouffe 1995, p. 22). The prime factorizations and distinct prime factors of the first few positive integers are listed in the table below.prime factorizationdistinct prime factors (A027748)1--0--221233134125515622, 377178129131022, 511111111222, 313131131422, 71523, 51612The numbers consisting only of distinct..

A sphenic number is a positive integer which is the product of exactly three distinct primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, ... (OEIS A007304).In particular, if , , and are prime numbers, then every sphenic number has precisely eight positive divisors, namely , , , , , , , and itself.The Möbius function of a sphenic number is .

Direct search factorization is the simplest (and most simple-minded) prime factorization algorithm. It consists of searching for factors of a number by systematically performing trial divisions, usually using a sequence of increasing numbers. Multiples of small primes are commonly excluded to reduce the number of trial divisors, but just including them is sometimes faster than the time required to exclude them. Direct search factorization is very inefficient, and can be used only with fairly small numbers.When using this method on a number , only divisors up to (where is the floor function) need to be tested. This is true since if all integers less than this had been tried, then(1)In other words, all possible factors have had their cofactors already tested. It is also true that, when the smallest prime factor of is , then its cofactor (such that ) must be prime. To prove this, suppose that the smallest is . If , then the smallest value and could assume..

Finch (2001, 2003) defines a -rough (or -jagged) number to be positive integer all of whose prime factors are greater than or equal to .Greene and Knuth define "unusual numbers" as numbers whose greatest prime factor is greater than or equal to , and these number are dubbed "-rough" or "-jagged" by Finch (2001, 2003). The first few unusual numbers are 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (OEIS A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). The first few "usual" numbers are then 8, 12, 16, 18, 24, 27, 30, ... (OEIS A063539).The probability that the greatest prime factor of a random integer is greater than is (Schroeppel 1972).

For an integer , let denote the greatest prime factor of , i.e., the number in the factorizationwith for . For , 3, ..., the first few are 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, ... (OEIS A006530). The greatest multiple prime factors for squareful integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, ... (OEIS A046028).A number for which is called an unusual number by Greene and Knuth (1990) and a -rough numbers by Finch (2001). The first few -rough numbers are 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, ... (OEIS A064052). The probability that a random positive integer is -rough is (Schroeppel 1972).A number that is not -rough is called, not surprisingly, a -smooth number (or sometimes, a "round number"). The first few are 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, ... (OEIS A048098)...

A prime factorization algorithm which uses residues produced in the continued fraction of for some suitably chosen to obtain a square number. The algorithm solvesby finding an for which (mod ) has the smallest upper bound. The method requires (by conjecture) about steps, and was the fastest prime factorization algorithm in use before the quadratic sieve, which eliminates the 2 under the square root (Pomerance 1996), was developed.

The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. 2-3).This theorem is also called the unique factorization theorem. The fundamental theorem of arithmetic is a corollary of the first of Euclid's theorems (Hardy and Wright 1979).For rings more general than the complex polynomials , there does not necessarily exist a unique factorization. However, a principal ideal domain is a structure for which the proof of the unique factorization property is sufficiently easy while being quite general and common.

The composite number problem asks if for a given positive integer there exist positive integers and such that .The complexity of the composite number problem was unknown for many years, although the problem was known to belong to (Pratt 1975, Garey and Johnson 1983). Agrawal et al. (2004) subsequently and unexpectedly discovered a polynomial-time algorithm now known as the AKS primality test.

Given an integer sequence , a prime number is said to be a primitive prime factor of the term if divides but does not divide any for . It is possible for a term to have zero, one, or many primitive prime factors.For example, the prime factors of the sequence are summarized in the following table (OEIS A005529).prime factorizationprime factorsprimitive prime factors12222255553102, 54171717175262, 13136373737377502, 58655, 139822, 414110101101101101

The prime signature of a positive integer is a sorted list of nonzero exponents in the prime factorizationBy definition, the prime signature of 1 is . The prime numbers have prime signature and squares of prime numbers have prime signature .The following table gives the prime signatures of the first few positive integers(OEIS A118914). factorizationprime signaturefactorizationprime signature11111122123313134145515616771717818919191020

Given a number , Fermat's factorization methods look for integers and such that . Then(1)and is factored. A modified form of this observation leads to Dixon's factorization method and the quadratic sieve.Every positive odd integer can be represented in the form by writing (with ) and noting that this gives(2)(3)Adding and subtracting,(4)(5)so solving for and gives(6)(7)Therefore,(8)As the first trial for , try , where is the ceiling function. Then check if(9)is a square number. There are only 22 combinations of the last two digits which a square number can assume, so most combinations can be eliminated. If is not a square number, then try(10)so(11)(12)(13)(14)Continue with(15)(16)(17)(18)(19)so subsequent differences are obtained simply by adding two.Maurice Kraitchik sped up the algorithm by looking for and satisfying(20)i.e., . This congruence has uninteresting solutions and interesting solutions . It turns out that if is odd..

The second part of Pollard rho factorization method concerns detection of the fact that a sequence has become periodic. Pollard's original suggestion was to use the idea attributed to Floyd of comparing to for all . Brent's improvement to Pollard's method concerns how to detect periodicity, and replaces Floyd's method with the following algorithm. Keep only one running copy of . If is a power of a base , let , and at each step, compare the current value with the saved value . In the factorization case, instead of comparing with , computeMore generally, Brent (1980) considered using any base for saving values instead of . However, he found to be very close to optimal.

A prime factor is a factor that is prime, i.e., one that cannot itself be factored. In general, a prime factorization takes the form(1)where are prime factors and are their orders. Prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.The following table gives the prime factorization for the positive integers .1111112131314141221222324233131323233343434142434445515253545616263646771717273737474781828384891919292939491020304050The number of not necessarily distinct prime factors of a number is denoted (Hardy and Wright 1979, p. 354) or . Conway et al. (2008) coined the term "multiprimality of " to describe , with semiprimes then being termed biprimes, numbers with three factors terms triprimes, etc. The number of prime factors is given in terms of the prime factorization above by(2)The first few values for , 2, ... are 0, 1, 1, 2, 1, 2,..

A factorization of the form(1)The factorization for was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as and as follows:(2)(3)which can be written(4)(5)(6)where and(7)(8)(9)

The Pratt certificate is a primality certificate based on Fermat's little theorem converse. Prior to the work of Pratt (1975), the Lucas-Lehmer test had been regarded purely as a heuristic that worked a lot of the time (Knuth 1969). Pratt (1975) showed that Lehmer's primality heuristic could be made a nondeterministic procedure by applying it recursively to the factors of . As a consequence of this result, Pratt (1975) became the first to demonstrate that the resulting tree implies that prime factorization lies in the complexity class NP.To generate a Pratt certificate, assume that is a positive integer and is the set of prime factors of . Suppose there exists an integer (called a "witness") such that but (mod ) whenever is one of . Then Fermat's little theorem converse states that is prime (Wagon 1991, pp. 278-279).By applying Fermat's little theorem converse to and recursively to each purported factor of , a certificate for..

A number with prime factorizationis called -almost prime if it has a sum of exponents , i.e., when the prime factor (multiprimality) function .The set of -almost primes is denoted .The primes correspond to the "1-almost prime" numbers and the 2-almost prime numbers correspond to semiprimes. Conway et al. (2008) propose calling these numbers primes, biprimes, triprimes, and so on.Formulas for the number of -almost primes less than or equal to are given byand so on, where is the prime counting function and is the th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; the first of which was discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006).The following table summarizes the first few -almost primes for small .OEIS-almost primes1A0000402, 3, 5, 7, 11, 13, ...2A0013584, 6, 9, 10, 14, 15, 21, 22, ...3A0146128, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52,..

A prime constellation, also called a prime -tuple, prime -tuplet, or prime cluster, is a sequence of consecutive numbers such that the difference between the first and last is, in some sense, the least possible. More precisely, a prime -tuplet is a sequence of consecutive primes (, , ..., ) with , where is the smallest number for which there exist integers , and, for every prime , not all the residues modulo are represented by , , ..., (Forbes). For each , this definition excludes a finite number of clusters at the beginning of the prime number sequence. For example, (97, 101, 103, 107, 109) satisfies the conditions of the definition of a prime 5-tuplet, but (3, 5, 7, 11, 13) does not because all three residues modulo 3 are represented (Forbes).A prime double with is of the form (, ) and is called a pair of twin primes. Prime doubles of the form (, ) are called cousin primes, and prime doubles of the form (, ) are called sexy primes.A prime triplet has . The constellation..

Given an arithmetic progression of terms , for , 2, ..., the series contains an infinite number of primes if and are relatively prime, i.e., . This result had been conjectured by Gauss (Derbyshire 2004, p. 96), but was first proved by Dirichlet (1837).Dirichlet proved this theorem using Dirichlet L-series, but the proof is challenging enough that, in their classic text on number theory, the usually explicit Hardy and Wright (1979) report "this theorem is too difficult for insertion in this book."

A lucky number of Euler is a number such that the prime-generating polynomialis prime for , 2, ..., . Such numbers are related to the imaginary quadratic field in which the ring of integers is factorable. Specifically, the lucky numbers of Euler (excluding the trivial case ) are those numbers such that the imaginary quadratic field has class number 1 (Rabinowitz 1913, Le Lionnais 1983, Conway and Guy 1996, Ribenboim 2000).As proved by Heegner (1952)--although his proof was not accepted as complete at the time--and subsequently established by Stark (1967), there are only nine numbers such that (the Heegner numbers , , , , , , , and ), and of these, only 7, 11, 19, 43, 67, and 163 are of the required form. Therefore, the only lucky numbers of Euler are 2, 3, 5, 11, 17, and 41 (Le Lionnais 1983, OEIS A014556), and there does not exist a better prime-generating polynomial of Euler's form...

A twin prime cluster of order is a collection of consecutive prime numbers such that consecutive pairs form twin primes. Twin prime clusters were discussed by Mudge (around 1995), and N. D. Backhouse found such a cluster of order 7. The smallest twin prime clusters of order , 2, ... are 3, 5, 5, 9419, 909287, 325267931, 678771479, 1107819732821, 170669145704411, 3324648277099157, ... (OEIS A111950). The cluster of order 7 was found by P. Carmody on Jan. 7, 2001, the cluster of order 8 was found by DeVries (2001), the cluster of order 9 by DeVries (2002), and the cluster of order 10 by G. Levai (in Sept. 2004, pers. comm., Apr. 5, 2005).

An odd prime is called a cluster prime if every even positive integer less than can be written as a difference of two primes , where . The first 23 odd primes 3, 5, 7, ..., 89 are all cluster primes. The first few odd primes that are not cluster primes are 97, 127, 149, 191, 211, ... (OEIS A038133).The numbers of cluster primes less than , , ... are 23, 99, 420, 1807, ... (OEIS A039506), and the corresponding numbers of noncluster primes are 0, 1, 68, 808, 7784, ... (OEIS A039507). It is not known if there are infinitely many cluster primes, but Blecksmith et al. (1999) show that for every positive integer , there is a bound such that if , thenwhere is the number of cluster primes not exceeding . Blecksmith et al. (1999) also show that the sum of the reciprocals of the cluster primes is finite...

A prime constellation of four successive primes with minimal distance . The term was coined by Paul Stäckel (1892-1919; Tietze 1965, p. 19). The quadruplet (2, 3, 5, 7) has smaller minimal distance, but it is an exceptional special case. With the exception of (5, 7, 11, 13), a prime quadruple must be of the form (, , , ). The first few values of which give prime quadruples are , 3, 6, 27, 49, 62, 69, 108, 115, ... (OEIS A014561), and the first few values of are 5 (the exceptional case), 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, ... (OEIS A007530). The number of prime quadruplets with largest member less than , , ..., are 1, 2, 5, 12, 38, 166, 899, 4768, ... (OEIS A050258; Nicely 1999).The asymptotic formula for the frequency of prime quadruplesis analogous to that for other prime constellations,(1)(2)where (OEIS A061642) is the Hardy-Littlewood constant for prime quadruplets.Roonguthai found the large prime quadruplets with(3)(4)(5)(6)(7)(8)(9)Forbes..

Let be an odd prime, be an integer such that and , andThen the following are equivalent 1. is prime. 2. There exists an such that , where GCD is the greatest common divisor (i.e., and are relatively prime). This is a modified version of the original theorem due to Lehmer.

Let be an odd integer, and assume there exists a Lucas sequence with associated Sylvester cyclotomic numbers such that there is an (with and relatively prime) for which divides . Then is a prime unless it has one of the following two forms: 1. , with prime and , or 2. , with and prime.

A test for the primality of Fermat numbers , with and . Then the two following conditions are equivalent: 1. is prime and , where is the Jacobi symbol, 2. . is usually taken as 3 as a first test.

A compositeness certificate is a piece of information which guarantees that a given number is composite. Possible certificates consist of a factor of a number (which, in general, is much quicker to check by direct division than to determine initially), or of the determination that either(i.e., violates Fermat's little theorem), orA quantity satisfying either property is said to be a witness to 's compositeness.

If a number fails Miller's primality test for some base , it is not a prime. If the number passes, it may be a prime. A composite number passing Miller's test is called a strong pseudoprime to base . If a number does not pass the test, then it is called a witness for the compositeness. If is an odd, positive composite number, then passes Miller's test for at most bases with (Long 1995). There is no analog of Carmichael numbers for strong pseudoprimes.The smallest numbers that are strong pseudoprimes to base 2, 3, 5, and 7 (and would hence fail a test based on these bases) are 3215031751, 118670087467, 307768373641, 315962312077, ... (OEIS A074773; Jaeschke 1993).Miller showed that any composite has a witness less than if the Riemann hypothesis is true.

For with odd and , if there exists an integer such thatthen is prime. A prime of this form is known as a Proth prime.

In August 2002, M. Agrawal and colleagues announced a deterministic algorithm for determining if a number is prime that runs in polynomial time (Agrawal et al. 2004). While this had long been believed possible (Wagon 1991), no one had previously been able to produce an explicit polynomial time deterministic algorithm (although probabilistic algorithms were known that seem to run in polynomial time). This test is now known as the Agrawal-Kayal-Saxena primality test, cyclotomic AKS test, or AKS primality test.Commenting on the impact of this discovery, P. Leyland noted, "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked" (quoted by Crandall and Papadopoulos 2003).The complexity of the original algorithm of Agrawal et al...

A primality test is a test to determine whether or not a given number is prime, as opposed to actually decomposing the number into its constituent prime factors (which is known as prime factorization).Primality tests come in two varieties: deterministic and probabilistic. Deterministic tests determine with absolute certainty whether a number is prime. Examples of deterministic tests include the Lucas-Lehmer test and elliptic curve primality proving. Probabilistic tests can potentially (although with very small probability) falsely identify a composite number as prime (although not vice versa). However, they are in general much faster than deterministic tests. Numbers that have passed a probabilistic prime test are therefore properly referred to as probable primes until their primality can be demonstrated deterministically.A number that passes a probabilistic test but is in fact composite is known as a pseudoprime. There are..

For an odd prime and a positive integer which is not a multiple of ,where is the Legendre symbol.

Let where is the factored part of a number(1)where , and .Pocklington's theorem, also known as the Pocklington-Lehmer test, then says that if there exists a for , ..., such that(2)and(3)then is prime.

There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.Group-theoretically, let be the modular group Gamma0, and let be the compactification (by adding cusps) of , where is the upper half-plane. Also define to be the Fricke involution defined by the block matrix . For a prime, define . Then is a supersingular prime if the genus of .The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field . They have their j-invariant in .Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group...

The unique even prime number 2. All other primes are odd primes. Humorously, that means 2 is the "oddest" prime of all.The sequence 2, 4, 6, 10, 14, 22, 26, 34, 38, ... (OEIS A001747) consisting of the number 2 together with the primes multiplied by 2 is sometimes also called the even primes, since these are the even numbers that are divisible by just 1, 2, , and .

A prime number is called circular if it remains prime after any cyclic permutation of its digits. An example in base-10 is because , , and are all primes. The first few circular primes are 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, ... (OEIS A068652).Base-10 circular primes not contain any digit 0, 2, 4, 5, 6, or 8, since having such a digit in the units place yields a number which is necessarily divisible by either or (and therefore not prime).Every prime repunit is a circular prime.Circular primes are rare. Including only the smallest number corresponding to each cycle gives the sequence 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, ... (OEIS A016114; Darling 2004), together with repunits , , , , , , and (the last several of which are probable primes)...

The summatory function of the totient function is defined by(1)(2)(3)(4)(Hardy and Wright 1979, p. 268), plotted as the red curve above. The first values of are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (OEIS A002088). has the asymptotic series(5)(6)where is the Riemann zeta function (Perrot 1881; Nagell 1951, p. 131; Hardy and Wright 1979, p. 268; blue curve above). An improved asymptotic estimate due to Walfisz (1963) is given by(7)Consider the summatory function of ,(8)plotted as the red curve above. For , 2, ..., the first few terms are 1, 2, 5/2, 3, 13/4, 15/4, 47/12, 25/6, ... (OEIS A028415 and A048049). The sum diverges as , but Landau (1900) showed that the asymptotic behavior is given by(9)where is the Euler-Mascheroni constant,(10)(11)(12)(13)(14)(15)(OEIS A082695), is the Möbius function, is the Riemann zeta function, and is the th prime (Landau 1900; Halberstam and Richert 1974, pp. 110-111; DeKoninck..

A totative is a positive integer less than or equal to a number which is also relatively prime to , where 1 is counted as being relatively prime to all numbers. The number of totatives of is the value of the totient function .The term was popularized by Sylvester (1879; Dickson 2005, p. 124), who spelled it "totitive."

The cototient of a positive number is defined as , where is the totient function. It is therefore the number of positive integers that have at least one prime factor in common with .The first few cototients for , 2, ... are 0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, ... (OEIS A051953).

Riemann defined the function by(1)(2)(3)(Hardy 1999, p. 30; Borwein et al. 2000; Havil 2003, pp. 189-191 and 196-197; Derbyshire 2004, p. 299), sometimes denoted , (Edwards 2001, pp. 22 and 33; Derbyshire 2004, p. 298), or (Havil 2003, p. 189). Note that this is not an infinite series since the terms become zero starting at , and where is the floor function and is the base-2 logarithm. For , 2, ..., the first few values are 0, 1, 2, 5/2, 7/2, 7/2, 9/2, 29/6, 16/3, 16/3, ... (OEIS A096624 and A096625). As can be seen, when is a prime, jumps by 1; when it is the square of a prime, it jumps by 1/2; when it is a cube of a prime, it jumps by 1/3; and so on (Derbyshire 2004, pp. 300-301), as illustrated above.Amazingly, the prime counting function is related to by the Möbius transform(4)where is the Möbius function (Riesel 1994, p. 49; Havil 2003, p. 196; Derbyshire 2004, p. 302). More amazingly..

The Legendre symbol is a number theoretic function which is defined to be equal to depending on whether is a quadratic residue modulo . The definition is sometimes generalized to have value 0 if ,(1)If is an odd prime, then the Jacobi symbol reduces to the Legendre symbol. The Legendre symbol is implemented in the Wolfram Language via the Jacobi symbol, JacobiSymbol[a, p].The Legendre symbol obeys the identity(2)Particular identities include(3)(4)(5)(6)(Nagell 1951, p. 144), as well as the general(7)when and are both odd primes.In general,(8)if is an odd prime.

The least common multiple of two numbers and , variously denoted (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), (Gellert et al. 1989, p. 25; Graham et al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West 2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325; Wolfram Language), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that(1)The least common multiple of more than two numbers is similarly defined.The least common multiple of , , ... is implemented in the Wolfram Language as LCM[a, b, ...].The least common multiple of two numbers and can be obtained by finding the prime factorization of each(2)(3)where the s are all prime factors of and , and if does not occur in one factorization, then the corresponding exponent is taken as 0. The least..

For a discrete function , the summatory function is defined bywhere is the domain of the function.

The odd part of a positive integer is defined bywhere is the exponent of the exact power of 2 dividing . is therefore the product of odd factors of . The values for , 2, ..., are 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... (OEIS A000265).

For a real number , the mantissa is defined as the positive fractional part , where denotes the floor function.For example, for , the mantissa is 0.14159.

Let be a mathematical statement, then the Iverson bracket is defined by(1)and corresponds to the so-called characteristic function. This notation conflicts with the brackets sometimes used to denote the floor function. (However, because of the elegant symmetry of the floor function and ceiling function symbols and , the use of to denote the floor function should be deprecated.)The Iverson bracket is implemented as a built-in function in the WolframLanguage as Boole[S].

For , let and be integers with such that the Euclidean algorithm applied to and requires exactly division steps and such that is as small as possible satisfying these conditions. Then and , where is a Fibonacci number (Knuth 1998, p. 343).Furthermore, the number of steps in the Euclidean algorithm never exceeds 5 times the number of digits in the smaller number. In fact, the bound 5 can be further reduced to , where is the golden ratio.

There are two different statements, each separately known as the greatest common divisor theorem.1. Given positive integers and , it is possible to choose integers and such that , where is the greatest common divisor of and (Eynden 2001). 2. If and are relatively prime positive integers, then there exist positive integers and such that (Johnson 1965).

The extended greatest common divisor of two integers and can be defined as the greatest common divisor of and which also satisfies the constraint for and given integers. It is used in solving linear Diophantine equations, and is implemented in the Wolfram Language as ExtendedGCD[m, n].

If and are integers not both equal to 0, then there exist integers and such thatwhere is the greatest common divisor of and .

The Mangoldt function is the function defined by(1)sometimes also called the lambda function. has the explicit representation(2)where denotes the least common multiple. The first few values of for , 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).The Mangoldt function is implemented in the WolframLanguage as MangoldtLambda[n].It satisfies the divisor sums(3)(4)(5)(6)where is the Möbius function (Hardy and Wright 1979, p. 254).The Mangoldt function is related to the Riemann zeta function by(7)where (Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).The summatory Mangoldt function, illustratedabove, is defined by(8)where is the Mangoldt function, and is also known as the second Chebyshev function (Edwards 2001, p. 51). is given by the so-called explicit formula(9)for and not a prime or prime power (Edwards 2001, pp. 49, 51, and 53), and the sum is over all nontrivial..

Let be an order of an imaginary quadratic field. The class equation of is the equation , where is the extension field minimal polynomial of over , with the -invariant of . (If has generator , then . The degree of is equal to the class number of the field of fractions of .The polynomial is also called the class equation of (e.g., Cox 1997, p. 293).It is also true thatwhere the product is over representatives of each ideal class of .If has discriminant , then the notation is used. If is not divisible by 3, the constant term of is a perfect cube. The table below lists the first few class equations as well as the corresponding values of , with being generators of ideals in each ideal class of . In each case, the constant term is written out as a cube times a cubefree part.0..

Let be a positive number, and define(1)(2)where the sum extends over the divisors of , and is the Möbius function. Then(3)(Nagell 1951, p. 286).For , 2, ..., is given by 0, 1, 3, 7, 11, 15, 20, 25, ... (OEIS A109507), where is the nearest integer function

The function(1)where is the number of not necessarily distinct prime factors of , with . The values of for , 2, ... are 1, , , 1, , 1, , , 1, 1, , , ... (OEIS A008836). The values of such that are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, ... (OEIS A026424), while then values such that are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).The Liouville function is implemented in the WolframLanguage as LiouvilleLambda[n].The Liouville function is connected with the Riemannzeta function by the equation(2)(Lehman 1960). It has the Lambert series(3)(4)where is a Jacobi theta function.Consider the summatory function(5)the values of which for , 2, ... are 1, 0, , 0, , 0, , , , 0, , , , , , 0, , , , , ... (OEIS A002819).Lehman (1960) gives the formulas(6)and(7)where , , and are variables ranging over the positive integers, is the Möbius function, is Mertens function, and , , and are positive real numbers with .The conjecture that satisfies for is called the..

The term "left factorial" is sometimes used to refer to the subfactorial , the first few values for , 2, ... are 1, 3, 9, 33, 153, 873, 5913, ... (OEIS A007489).Unfortunately, the same term and notation are also applied to the factorialsum(1)(2)(3)(4)where is a gamma function, is the exponential integral, and is the En-function.For , 1, ..., the first few values are given by 0, 1, 2, 4, 10, 34, 154, 874, ... (OEIS A003422). The left factorial is always even for . is prime for , 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, ... (OEIS A100614), the last of which was found by E. W. Weisstein (Oct. 19, 2006).

If and are relatively prime so that the greatest common divisor , thenwhere is the Carmichael function.

A multiplicative number theoretic function is a number theoretic function that has the property(1)for all pairs of relatively prime positive integers and .If(2)is the prime factorization of a number , then(3)Multiplicative number theoretic functions satisfy the amazing identity(4)(5)where the product is over the primes.

There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer such that for all relatively prime to . The multiplicative order of (mod ) is at most (Ribenboim 1989). The first few values of this function, implemented as CarmichaelLambda[n], are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, ... (OEIS A002322).It is given by the formula(1)where are primaries.It can be defined recursively as(2)Some special values include(3)and(4)where is a primorial (S. M. Ruiz, pers. comm., Jul. 5, 2009).The second Carmichael's function is given by the least common multiple (LCM) of all the factors of the totient function , except that if , then is a factor instead of . The values of for the first few are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, ... (OEIS A011773).This function has the special value(5)for an odd prime and ...

The so-called explicit formulagives an explicit relation between prime numbers and Riemann zeta function zeros for and not a prime or prime power. Here, is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros of the Riemann zeta function , i.e., those in the critical strip so (Montgomery 2001).

The Möbius function is a number theoretic function defined by(1)so indicates that is squarefree (Havil 2003, p. 208). The first few values of are therefore 1, , , 0, , 1, , 0, 0, 1, , 0, ... (OEIS A008683). Similarly, the first few values of for , 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).The function was introduced by Möbius (1832), and the notation was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number ] is either (when is divisible by a square), or (mod ) (when is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).The Möbius function is implemented in the WolframLanguage as MoebiusMu[n].The summatory function of the Möbius function(2)is called the Mertens function.The..

Given relatively prime integers and (i.e., ), the Dedekind sum is defined by(1)where(2)with the floor function. is an odd function since and is periodic with period 1. The Dedekind sum is meaningful even if , so the relatively prime restriction is sometimes dropped (Apostol 1997, p. 72). The symbol is sometimes used instead of (Beck 2000).The Dedekind sum can also be expressed in the form(3)If , let , , ..., denote the remainders in the Euclidean algorithm given by(4)(5)(6)for and . Then(7)(Apostol 1997, pp. 72-73).In general, there is no simple formula for closed-form evaluation of , but some special cases are(8)(9)(Apostol 1997, p. 62). Apostol (1997, p. 73) gives the additional special cases(10)(11)(12)(13)for and , where and . Finally,(14)for and , where or .Dedekind sums obey 2-term(15)(Dedekind 1953; Rademacher and Grosswald 1972; Pommersheim 1993; Apostol 1997, pp. 62-64) and 3-term(16)(Rademacher..

Define(1)where(2)(Davenport 1980, p. 121), is the Mangoldt function, and is the totient function. Now define(3)where the sum is over relatively prime to , , and(4)Bombieri's theorem then says that for fixed ,(5)provided that .

The Mertens function is the summary function(1)where is the Möbius function (Mertens 1897; Havil 2003, p. 208). The first few values are 1, 0, , , , , , , , , , , ... (OEIS A002321). is also given by the determinant of the Redheffer matrix.Values of for , 1, 2, ... are given by 1, , 1, 2, , , 212, 1037, 1928, , ... (OEIS A084237; Deléglise and Rivat 1996).The following table summarizes the first few values of at which for various OEIS such that 13, 19, 20, 30, 33, 43, 44, 45, 47, 48, 49, 50, ...5, 7, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 25, 29, ...3, 4, 6, 10, 15, 16, 22, 26, 27, 28, 35, 36, 38, ...0A0284422, 39, 40, 58, 65, 93, 101, 145, 149, 150, ...1A1186841, 94, 97, 98, 99, 100, 146, 147, 148, 161, ...295, 96, 217, 229, 335, 336, 339, 340, 345, 347, 348, ...3218, 223, 224, 225, 227, 228, 341, 342, 343, 344, 346, ...An analytic formula for is not known, although Titchmarsh (1960) showed that if the Riemann hypothesis holds and if there are no multiple Riemann..

The Dedekind -function is defined by the divisor product(1)where the product is over the distinct prime factors of , with the special case . The first few values are(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)giving 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, ... (OEIS A001615).Sums for include(12)(13)where is the Möbius function.The Dirichlet generating functionis given by(14)(15)where is the Riemann zeta function.

The sum of the aliquot divisors of , given bywhere is the divisor function. The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (OEIS A001065).

A number is said to be refactorable, sometimes also called a tau number (Kennedy and Cooper 1990), if it is divisible by the number of its divisors , where is the divisor function.The first few refactorable numbers are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, ...(OEIS A033950).The first new such that and are both refactorable numbers are 1, 8, 1520, 50624, 62000, 103040, ... (OEIS A114617).Zelinsky (2002) proved that there are no refactorable numbers and such that and also Colton's conjecture that no three consecutive integers can all be refactorable.

Given an amicable pair , the quantity(1)(2)(3)is called the pair sum, where is the divisor function and is the restricted divisor function.

The odd divisor function(1)is the sum of th powers of the odd divisors of a number . It is the analog of the divisor function for odd divisors only.For the case ,(2)(3)(4)where is defined to be 0 if is odd. The generating function is given by(5)(6)(7)where is a Jacobi elliptic function.Rather surprisingly, gives the number of factors of the polynomial .The following table gives the first few .OEIS0A0012271, 1, 2, 1, 2, 2, 2, 1, 3, 2, ...1A0005931, 1, 4, 1, 6, 4, 8, 1, 13, 6, ...2A0509991, 1, 10, 1, 26, 10, 50, 1, 91, 26, ...3A0510001, 1, 28, 1, 126, 28, 344, 1, 757, 126, ...4A0510011, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, ...5A0510021, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, ...This function arises in Ramanujan's Eisenstein series and in a recurrence relation for the partition function P...

Let be the divisor function. Thenwhere is the Euler-Mascheroni constant. Ramanujan independently discovered a less precise version of this theorem (Berndt 1985).

The sum of powers of even divisors of a number. It is the analog of the divisor function for even divisors only and is written . It is given simply in terms of the usual divisor function by(1)

By analogy with the divisor function , let(1)denote the product of the divisors of (including itself). For , 2, ..., the first few values are 1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, ... (OEIS A007955).The divisor product satisfies the identity(2)The following table gives values of for which is a th power. Lionnet (1879) considered the case .OEIS2A0489431, 6, 8, 10, 14, 15, 16, 21, 22, 24, 26, ...3A0489441, 4, 8, 9, 12, 18, 20, 25, 27, 28, 32, ...4A0489451, 24, 30, 40, 42, 54, 56, 66, 70, 78, ...5A0489461, 16, 32, 48, 80, 81, 112, 144, 162, ...Write the prime factorization of a number ,(3)Then the power of occurring in is(4)(Kaplansky 1999). This allows rules for determining when is a power of to be determined, as considered by Halcke (1719) and Lionnet (1879). Let , , and be distinct primes, then the following table gives the conditions and first few for which is a given power of (Ireland and Rosen 1990, Kaplansky 1999, Dickson 2005). The case of third..

The divisor function for an integer is defined as the sum of the th powers of the (positive integer) divisors of ,(1)It is implemented in the Wolfram Language as DivisorSigma[k, n].The notations (Hardy and Wright 1979, p. 239), (Ore 1988, p. 86), and (Burton 1989, p. 128) are sometimes used for , which gives the number of divisors of . Rather surprisingly, the number of factors of the polynomial are also given by . The values of can be found as the inverse Möbius transform of 1, 1, 1, ... (Sloane and Plouffe 1995, p. 22). Heath-Brown (1984) proved that infinitely often. The numbers having the incrementally largest number of divisors are called highly composite numbers. The function satisfies the identities(2)(3)where the are distinct primes and is the prime factorization of a number .The divisor function is odd iff is a square number.The function that gives the sum of the divisors of is commonly written without the..

Let the divisor function be the number of divisors of (including itself). For a prime , . In general,where is the Euler-Mascheroni constant. Dirichlet originally gave (Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68), and Hardy and Landau showed in 1916 that (Hardy 1999, p. 81). The following table summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).approx.citation1/20.50000Dirichlet1/30.33333Voronoi (1903), Sierpiński (1906), van der Corput (1923)37/1120.33036Littlewood and Walfisz (1925)33/1000.33000van der Corput (1922)27/820.32927van der Corput (1928)15/460.3260912/370.32432Chen (1963), Kolesnik (1969)35/1080.32407Kolesnik (1982)139/4290.32401Kolesnik17/530.32075Vinogradov (1935)7/220.31818Iwaniec and Mozzochi (1988)23/730.31507Huxley (1993)131/4160.31490Huxley (2003)..

A number theoretic character, also called a Dirichlet character (because Dirichlet first introduced them in his famous proof that every arithmetic progression with relatively prime initial term and common difference contains infinitely many primes), modulo is a complex function for positive integer such that(1)(2)(3)for all , and(4)if . can only assume values which are roots of unity, where is the totient function.Number theoretic characters are implemented in the Wolfram Language as DirichletCharacter[k, j, n], where is the modulus and is the index.

A continuous homomorphism of a group into the nonzero complex numbers. A multiplicative character gives a group representation on the one-dimensional space of complex numbers, where the group representation action by is multiplication by . A multiplicative character is unitary if it has absolute value 1 everywhere.

The Jacobi symbol as a number theoretic character can be extended to the Kronecker symbol so that whenever . When is relatively prime to , then , and for nonzero values iff . In addition, is the minimum value for which the latter congruence property holds in any extension symbol for .

The Dirichlet kernel is obtained by integrating the number theoretic character over the ball ,

An integer sequence whose terms are defined in terms of number-related words in some language. For example, the following table gives the sequences of numbers having digits whose English names (zero, one, two, three, four, five, six, seven, eight, nine) are in alphabetical order and also satisfy some other property.propertyOEISsequenceorderedA0534321, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...distinct, orderedA0534331, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, ...prime, orderedA0534342, 3, 5, 7, 11, 13, 17, 41, 43, 47, 53, 59, ...distinct, prime, orderedA0534352, 3, 5, 7, 13, 17, 41, 43, 47, 53, 59, 73, ...

By way of analogy with the eban numbers, uban numbers are defined as numbers whose English names do not contain the letter "u" (i.e., "u" is banned). Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of such numbers.The first few uban numbers are 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, ... (OEIS A089590). The sequence of uban numbers first differs from OEIS A052406 (the numbers not containing the digit 4) at the term 40 (forty), which is a uban number but is not 4-less.A plot of the first few uban numbers represented as a sequence of binary bits is shown above. The top portion shows the first 255 values, and the bottom shows the next 510 values...

By way of analogy with the eban numbers, oban numbers are defined as numbers whose English names do not contain the letter "o" (i.e., "o" is banned). Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of large numbers.The first few are 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 23, 25,26, ... (OEIS A008521).Since the number name for every power of 10 greater than 3 contains either "thousand" or the suffix "-illion", there are a finite number of oban numbers. In fact, there are a total of 454 of them, the largest of which is 999.A plot of the oban numbers represented as a sequence of binary bits is shown above.

The integer sequence beginning with a single digit in which the next term is obtained by describing the previous term. Starting with 1, the sequence would be defined by "1, one 1, two 1s, one 2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, .... Similarly, starting the sequence instead with the digit for gives , 1, 111, 311, 13211, 111312211, 31131122211, 1321132132211, ..., as summarized in the following table.OEISsequence1A0051501, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...2A0067512, 12, 1112, 3112, 132112, 1113122112, 311311222112, ...3A0067153, 13, 1113, 3113, 132113, 1113122113, 311311222113, ...The number of digits in the th term of the sequence for are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, ... (OEIS A005341). Similarly, the numbers of digits for the th term of the sequence for , 3, ..., are 1, 2, 4, 4, 6, 10, 12, 14, 22, 26, ... (OEIS A022471). These sequences are asymptotic to , where(1)(2)(3)The quantity..

By way of analogy with the eban numbers, iban numbers are defined as numbers whose English names do not contain the letter "i" (i.e., "i" is banned). The first few are 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, ... (OEIS A089589). Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of such numbers.Ignoring "special" number names such as googol and googolplex, the number name for every power of 10 greater than 5 contains the suffix "-illion," so there are a finite number of iban numbers. In fact, there are a total of 30275 of them, the largest of which is 777777.A plot of the first few iban numbers represented as a sequence of binary bits is shown above. The top portion shows the first 255 values, and the bottom shows the next 510 values...

The eban numbers are the sequence of numbers whose names (in English) do not contain the letter "e" (i.e., "e" is "banned"). The name was coined by N. J. A. Sloane around 1990. Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of such numbers.The first few eban numbers are 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, ... (OEIS A006933); i.e., two, four, six, thirty, etc. These exclude one, three, five, seven, eight, nine, ten, eleven, twelve, etc.In English, every odd number contains an "e," so all eban numbers are even (Hernandez et al. 2002-2003). In addition, eban numbers satisfy the following properties (Hernandez et al. 2002-2003). 1. There are gaps larger than any given number between eban numbers. 2. If a number of the form is an eban..

There exists an integer such that every string in the look and say sequence "decays" in at most days to a compound of "common" and "transuranic elements."The table below gives the periodic table of atoms associated with the look and say sequence as named by Conway (1987). The "abundance" is the average number of occurrences for long strings out of every million atoms. The asymptotic abundances are zero for transuranic elements, and 27.246... for arsenic (As), the next rarest element. The most common element is hydrogen (H), having an abundance of . The starting element is U, represented by the string "3," and subsequent terms are those giving a description of the current term: one three (13); one one, one three (1113); three ones, one three (3113), etc.abundance is the derivate of 102.5628524992U39883.598639291Pa137581.904712590Th11136926.935204589Ac31135313.789499988Ra1321134076.313407887Fr11131221133127.020932886Rn3113112221132398.799831185AtHo.13221131840.166968384Po11132221131411.628610083Bi31133221131082.888328582PbPm.123222113830.7051329381Tl111213322113637.2503975580Hg31121123222113488.8474298279Au132112211213322113375.0045673878Pt111312212221121123222113287.6734477577Ir3113112211322112211213322113220.6800122976Os1321132122211322212221121123222113169.2880180875Re111312211312113221133211322112211213322113315.5665525274WGe.Ca.312211322212221121123222113242.0773666673Ta131122211332113221122112133221132669.097036372Hf11132.Pa.H.Ca.W2047.517320071Lu3113121570.691180870Yb13211311121204.908384169Tm111312211331121098.595599768Er311311222.Ca.Co47987.52943867Ho1321132.Pm36812.18641866Dy11131221131228239.35894965Tb311311222113111221662.97282164GdHo.1322113311220085.66870963Eu1113222.Ca.Co15408.11518262Sm31133229820.45616761Pm132.Ca.Zn22875.86388360Nd11131217548.52928759Pr3113111213461.82516658Ce132113311210326.83331257La11131.H.Ca.Co7921.918828456Ba3113116077.061188955Cs132113214661.834272054Xe111312211312113576.185610753I3113112221131112212743.362971852TeHo.13221133122112104.488193351SbEu.Ca.31122211614.394668750SnPm.132111238.434197249In11131221950.0274564648Cd3113112211728.7849205647Ag132113212221559.0653794646Pd111312211312113211428.8701504145Rh311311222113111221131221328.9948057644RuHo.132211331222113112211386.0770494343TcEu.Ca.311322113212221296.1673685242Mo13211322211312113211227.1958675241Nb1113122113322113111221131221174.2864599740ZrEr.12322211331222113112211133.6986031539Y1112133.H.Ca.Tc102.5628524938Sr3112112.U78.67800008937Rb132112211260.35545568236Kr1113122122211246.29986815235Br311311221132211235.51754794434Se1321132122211322211227.24621607633As111312211312113221133221121887.437227632Ge31131122211311122113222.Na1447.890564231GaHo.1322113312221133223571.39133630ZnEu.Ca.Ac.H.Ca.31218082.08220329Cu13111213871.12320028Ni1113311245645.87725627CoZn.3211235015.85854626Fe1312211226861.36018025Mn11131122211220605.88261124Cr31132.Si15807.18159223V1321131212126.00278322Ti111312211311129302.097444321Sc311311222113311256072.54312920CaHo.Pa.H.12.Co43014.36091319K111232997.17012218Ar311225312.78421817Cl13211219417.93925016S111312211214895.88665815P31131122211232032.81296014SiHo.132211224573.00669613Al111322211218850.44122812Mg311332211214481.44877311NaPm.12322211211109.00669610Ne1112133221128521.93965399F311211232221126537.34907508O1321122112133221125014.93024647N1113122122211211232221123847.05254196C31131122113221122112133221122951.15037165B13211321222113222122211211232221122263.88603254Be1113122113121132211332113221122112133221124220.06659823LiGe.Ca.3122113222122211211232221123237.29685882He1311222113321132211221121332211291790.3832161HHf.Pa.22.Ca.Li..

By way of analogy with the eban numbers, aban numbers are defined as numbers whose English names do not contain the letter "a" (i.e., "a" is banned). Note that this definition is imprecise insofar as special names are sometimes assigned to a few large numbers that do not follow the usual rules for the naming of such numbers.Since the word "thousand" contains an "a" but no smaller numbers do, the numbers 1-999, -, -, ... etc. are aban numbers.

Let(1)(2)(3)(OEIS A104457), where is the golden ratio, and(4)(5)(OEIS A002390).Define the Fibonacci hyperbolic sine by(6)(7)(8)The function satisfies(9)and for ,(10)where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).Define the Fibonacci hyperbolic cosine by(11)(12)(13)This function satisfies(14)and for ,(15)where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).Similarly, the Fibonacci hyperbolic tangent is defined by(16)and for ,(17)For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).

Let be the sum of prime factors (with repetition) of a number . For example, , so . Then for , 2, ... is given by 0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, ... (OEIS A001414). The sum of prime factors function is also known as the integer logarithm.The high-water marks are 0, 2, 3, 4, 5, 7, 11, 13, 17, ..., which occur at positions 1, 2, 3, 4, 5, 7, 11, 13, 17, ... (OEIS A046022), which, with the exception of the first term, correspond exactly to the actual values of the high-water marks.If is considered to be 0 for a prime, then the sequence of high-water marks is 0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, ... (OEIS A088685), which occur at positions 1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, ... (OEIS A088686). Rather amazingly, if the first 7 terms are dropped, then the last digit of the high-water marks and the last digit of their positions fall into one of the four patterns , (3, 2), (5, 6), or (9, 4) (A. Jones, pers. comm., October 5, 2003).Now consider iterating..

Let , , and denote positive integers satisfying(i.e., both pairs are relatively prime), and suppose every prime with is expressible if the form for some integers and . Then every prime such that and is expressible in the form for some integers and (Halter-Koch 1993, Williams 1991).prime formrepresentation

Let(1)be the sum of the first primes (i.e., the sum analog of the primorial function). The first few terms are 2, 5, 10, 17, 28, 41, 58, 77, ... (OEIS A007504). Bach and Shallit (1996) show that(2)and provide a general technique for estimating such sums.The first few values of such that is prime are 1, 2, 4, 6, 12, 14, 60, 64, 96, 100, ... (OEIS A013916). The corresponding values of are 2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, ... (OEIS A013918).The first few values of such that are 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, ... (OEIS A045345). The corresponding values of are 2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, ... (OEIS A050247; Rivera), and the values of are 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (OEIS A050248; Rivera).In 1737, Euler showed that the harmonic seriesof primes, (i.e., sum of the reciprocals of the primes) diverges(3)(Nagell..

Fermat's theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number can be represented in an essentially unique manner (up to the order of addends) in the form for integer and iff or (which is a degenerate case with ). The theorem was stated by Fermat, but the first published proof was by Euler.The first few primes which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (OEIS A002313) (with the only prime congruent to 2 mod 4 being 2). The numbers such that equal these primes are (1, 1), (1, 2), (2, 3), (1, 4), (2, 5), (1, 6), ... (OEIS A002331 and A002330).The theorem can be restated by lettingthen all relatively prime solutions to the problem of representing for any integer are achieved by means of successive applications of the genus theorem and composition theorem...

For any , there exists a such that the sequencewhere , 2, ... contains at least primes.

Euler's theorem states that every prime of the form , (i.e., 7, 13, 19, 31, 37, 43, 61, 67, ..., which are also the primes of the form ; OEIS A002476) can be written in the form with and positive integers.The first few positive integers that can be represented in this form (with ) are 4, 7, 12, 13, 16, 19, ... (OEIS A092572), summarized in the following table together with their representations.4(1, 1)7(2, 1)12(3, 1)13(1, 2)16(2, 2)19(4, 1)21(3, 2)28(1, 3), (4, 2), (5, 1)31(2, 3)Restricting solutions such that (i.e., and are relatively prime), the numbers that can be represented as are 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, ... (OEIS A092574), as summarized in the following table. with 4(1, 1)7(2, 1)12(3, 1)13(1, 2)19(4, 1)21(3, 2)28(1, 3), (5, 1)31(2, 3)37(5, 2)

There exists a positive integer such that every sufficiently large integer is the sum of at most primes. It follows that there exists a positive integer such that every integer is a sum of at most primes. The smallest proven value of is known as the Schnirelmann constant.Schnirelmann's theorem can be proved using Mann's theorem,although Schnirelmann used the weaker inequalitywhere , , and is the Schnirelmann density. Let be the set of primes, together with 0 and 1, and let . Using a sophisticated version of the inclusion-exclusion principle, Schnirelmann showed that although , . By repeated applications of Mann's theorem, the sum of copies of satisfies . Thus, if , the sum of copies of has Schnirelmann density 1, and so contains all positive integers.

Erdős proved that there exist at least one prime of the form and at least one prime of the form between and for all .

Every "large" even number may be written as where is a prime and is the set of primes and semiprimes .

There are at least two theorems known as Chebyshev's theorem.The first is Bertrand's postulate, proposed by Bertrand in 1845 and proved by Chebyshev using elementary methods in 1850 (Derbyshire 2004, p. 124).The second is a weak form of the prime number theorem stating that the order of magnitude of the prime counting function iswhere denotes "is asymptotic to" (Hardy and Wright 1979, p. 9). More precisely, Chebyshev showed in 1849 that iffor some constant , then (Derbyshire 2004, p. 123).

A pair of prime numbers such thatThe only known examples are (2, 1093), (3, 1006003), (5 , 1645333507), (83, 4871), (911, 318917), and (2903, 18787).If the equation of Catalan's Diophantineproblemhas a nontrivial solution in integers and primes greater than 3, then must be a double Wieferich pair, as proved in 2000 by Mihailescu (Steiner 1998, Peterson 2000).

Mills' constant can be defined as the least such thatis prime for all positive integers (Caldwell and Cheng 2005).The first few for , 2, ... are 2, 11, 1361, 2521008887, ... (OEIS A051254). They can be represented more compactly through as andCaldwell and Cheng (2005) calculated the first 10 Mills primes. 13 are known as of Jul. 2013, with the firth few for , 2, ... being 3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, ... (OEIS A108739). is not known, but it is known that (E. Weisstein, Aug. 13, 2013).The integer lengths of the Mills' primes are 1, 2, 4, 10, 29, 85, 254, 762, 2285,6854, 20562, 61684, 185052, ... (OEIS A224845).

A number such that the "LED representation" of (i.e., the arrangement of horizonal and vertical lines seen on a digital clock or pocket calculator), upside down, in a mirror, and upside-down-and-in-a-mirror are all primes. The digits of are therefore restricted to 0, 1, 2, 5, and 8. The first few dihedral primes are 2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, ... (OEIS A134996).

An interprime is the average of consecutive (but not necessarily twin) odd primes. The first few terms are 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, ... (OEIS A024675). The first few even interprimes are 4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, ... (OEIS A072568), and the first few odd ones are 9, 15, 21, 39, 45, 69, 81, 93, 99, ... (OEIS A072569).Interprimes cannot themselves be prime (since otherwise there would exist a prime between consecutive primes, which is impossible by definition).The sumhas zeros at almost integer approximations of the interprimes, with the single additional point 5/2 (D. Tisdale, pers. comm., Sep. 8, 2008).

A prime is called "good" iffor all (there is a typo in Guy 1994 in which the s are replaced by 1s). There are infinitely many good primes, and the first few are 5, 11, 17, 29, 37, 41, 53, ... (OEIS A028388).

A prime which does not divide the class number of the cyclotomic field obtained by adjoining a primitive pth root of unity to the field of rationals. A prime is regular iff does not divide the numerators of the Bernoulli numbers , , ..., . A prime which is not regular is said to be an irregular prime.In 1915, Jensen proved that there are infinitely many irregular primes. It has not yet been proven that there are an infinite number of regular primes (Guy 1994, p. 145). Of the primes , (or 60.59%) are regular (the conjectured fraction is ). The first few are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (OEIS A007703).

A Proth number that is prime, i.e., a number of the form for odd , a positive integer, and . Factors of Fermat numbers are of this form as long as they satisfy the condition odd and . For example, the factor of is not a Proth prime since . (Otherwise, every odd prime would be a Proth prime.)Proth primes satisfy Proth's theorem, i.e., a number of this form is prime iff there exists a number a such that is congruent to modulo . This provides an easy computational test for Proth primes. Yves Gallot has written a downloadable program for testing Proth primes and many of the largest currently known primes have been found with this program.A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, ...(OEIS A080076).The following table gives the first few indices such that is prime for..

The quotientwhich must be congruent to 0 (mod ) for to be a Wilson prime. The quotient is an integer only when (in which case ) or is a prime, and the values of corresponding to , 3, 5, 7, 11, ... are 1, 1, 5, 103, 329891, 36846277, 1230752346353, ... (OEIS A007619).

Mann's theorem is a theorem widely circulated as the " conjecture" that was subsequently proven by Mann (1942). It states that if and are sets of integers each containing 0, thenHere, denotes the direct sum, i.e., , and is the Schnirelmann density.Mann's theorem is optimal in the sense that satisfies .Mann's theorem implies Schnirelmann's theorem as follows. Let , then Mann's theorem proves that , so as more and more copies of the primes are included, the Schnirelmann density increases at least linearly, and so reaches 1 with at most copies of the primes. Since the only sets with Schnirelmann density 1 are the sets containing all positive integers, Schnirelmann's theorem follows.

Let be an irregular prime, and let be a prime with . Also let be an integer such that (mod ). For an irregular pair , form the product(1)where(2)(3)If (mod ) for all such irregular pairs, then Fermat's last theorem holds for exponent .

Chebyshev noticed that the remainder upon dividing the primes by 4 gives 3 more often than 1, as plotted above in the left figure. Similarly, dividing the primes by 3 gives 2 more often than 1 (right figure). This is called the Chebyshev bias, or sometimes the prime race (Wagon 1994).Consider the list of the first primes (mod 4). This list contains equal numbers of remainders 3 and 1 (mod 4) for , 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, ... (OEIS A038691; Wagon 1994, pp. 2-3). The values of for which the list is biased towards 1 are 2946, 50378, 50380, 50382, 50383, 50384, 50385, ... (OEIS A096628).Definingthe values of for which are , 3, 7, 13, 89, 2943, 2945, 2947, ... (OEIS A038691).Similarly, consider the list of the first primes (mod 3), skipping and since . This list contains equal numbers of remainders 2 and 1 at the values , 6, 8, 12, 14, 22, 38, 48, 50, ... (OEIS A096629). The first value of for which the list is biased towards 1 is , as..

There are several types of numbers that are commonly termed "lucky numbers."The first is the lucky numbers of Euler. The second is obtained by writing out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is , just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.It therefore appears that the sieving process accountsfor many properties of the primes...

An Euler pseudoprime to the base is a composite number which satisfiesThe first few base-2 Euler pseudoprimes are 341, 561, 1105, 1729, 1905, 2047, ...(OEIS A006970).

A Poulet number is a Fermat pseudoprime to base 2, denoted psp(2), i.e., a composite number such thatThe first few Poulet numbers are 341, 561, 645, 1105, 1387, ... (OEIS A001567).Pomerance et al. (1980) computed all Poulet numbers less than . The numbers less than , , ..., are 0, 3, 22, 78, 245, ... (OEIS A055550).Pomerance has shown that the number of Poulet numbers less than for sufficiently large satisfy(Guy 1994).A Poulet number all of whose divisors satisfy is called a super-Poulet number. There are an infinite number of Poulet numbers which are not super-Poulet numbers. Shanks (1993) calls any integer satisfying (i.e., not limited to odd composite numbers) a Fermatian.

Two integers form a super unitary amicable pair ifwhere is the unitary divisor function. The first few pairs are (105, 155), (110, 142), (2145, 3055), (47802, 65278), (125460, 164492), ... (OEIS A045613 and A045614).

A number is said to be cubefree if its prime factorization contains no tripled factors. All primes are therefore trivially cubefree. The cubefree numbers are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ... (OEIS A004709). The cubeful numbers (i.e., those that contain at least one cube) are 8, 16, 24, 27, 32, 40, 48, 54, ... (OEIS A046099). The number of cubefree numbers less than 10, 100, 1000, ... are 9, 85, 833, 8319, 83190, 831910, ..., and their asymptotic density is , where is the Riemann zeta function.

Let and denote the number and sum of the divisors of , respectively (i.e., the zeroth- and first-order divisor functions). A number is called sublime if and are both perfect numbers. The only two known sublime numbers are 12 and(Math Pages). It is not known if any odd sublime numberexists.

A colossally abundant number is a positive integer for which there is a positive exponent such thatfor all . All colossally abundant numbers are superabundant numbers.The first few are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, ... (OEIS A004490). The following table lists the colossally abundant numbers up to , as given by Alaoglu and Erdős (1944).factorization of 221.50062.000122.333602.8001203.0003603.25025203.71450403.838554404.1877207204.50914414404.58143243204.699216216004.8553675672005.14169837768005.4121606268664005.6473212537328005.69293163582512005.8882888071057872006.07820216497405104006.18760649492215312006.2382244031211966544006.407The first 15 elements of this sequence agree with those of the superiorhighly composite numbers (OEIS A002201).The th colossally abundant number has the form , where..

Two integers and are -multiamicable ifandwhere is the divisor function and are positive integers. If , is an amicable pair. cannot have just one distinct prime factor, and if it has precisely two distinct prime factors, then and is even. Small multiamicable numbers for small are given by Cohen et al. (1995). Several of these numbers are reproduced in the table below.167645528818310219217529201522801716225560405802801790863136227249568171622556040580280177082132428817712480614417199615613902848499240550375424

The th Monica set is defined as the set of composite numbers for which , where(1)(2)and(3)(4)Every Monica set has an infinite number of elements.The Monica set is a superset of the Suzanne set .The following table gives the first few Monica numbers in for small .OEIS1A0182521, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...2A1022184, 8, 10, 12, 14, 15, 22, 26, 27, 35, 42, 44, ...3A1022199, 16, 24, 27, 28, 32, 40, 42, 49, 52, 56, 60, ...If is a Smith number, then it is a member of the Monica set for all . For any integer , if is a -Smith number, then .

A pair of positive integers such that the equations(1)have a positive integer solution , where is the divisor function. If is prime, then is an amicable pair (te Riele 1986). is a "special" breeder if(2)(3)where and are relatively prime, . If regular amicable pairs of type with are of the form with prime, then are special breeders (te Riele 1986).

Sociable numbers are numbers that result in a periodic aliquot sequence, where an aliquot sequence is the sequence of numbers obtained by repeatedly applying the restricted divisor function(1)to and is the usual divisor function.If the period of the aliquot cycle is 1, the number is called a perfect number. If the period is 2, the two numbers are called an amicable pair. In general, if the period is , the number is called sociable of order . For example, 1264460 is a sociable number of order four since its aliquot sequence is 1264460, 1547860, 1727636, 1305184, 1264460, ....Only two groups of sociable numbers were known prior to 1970, namely the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4.The first few sociable numbers are 12496, 14316, 1264460, 2115324, 2784580, 4938136, ... (OEIS A003416), which have orders 5, 28, 4, 4, 4, 4, ... (OEIS A052470). The following table summarizes the smallest..

An integer is -smooth if it has no prime factors . The following table gives the first few -smooth numbers for small . Berndt (1994, p. 52) called the 7-smooth numbers "highly composite numbers."OEIS-smooth numbers2A0000791, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...3A0035861, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ...5A0510371, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, ...7A0024731, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, ...11A0510381, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, ...The probability that a random positive integer is -smooth is , where is the number of -smooth numbers . This fact is important in application of Kraitchik's extension of Fermat's factorization method because it is related to the number of random numbers which must be examined to find a suitable subset whose product is a square.Since about -smooth numbers must be found (where is the prime counting function), the number of random numbers which must be examined is about . But because it..

Let be the sum of the infinitary divisors of a number . An infinitary perfect number is a number such that . The first few are 6, 60, 90, 36720, ... (OEIS A007357). Cohen (1990) found 14 such numbers, and 155 are known as of January 2004 (Pedersen).

An amicable quadruple as a quadruple such that(1)where is the divisor function.If and are amicable pairs and(2)then is an amicable quadruple. This follows from the identity(3)The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800).Large amicable quadruples can be generated using the formula(4)where(5)and is a Mersenne prime with a prime (Y. Kohmoto; Guy 1994, p. 59).

Let be the sum of the infinitary divisors of a number . An infinitary -multiperfect number is a number such that . Cohen (1990) found 13 infinitary 3-multiperfects, seven 4-multiperfects, and two 5-multiperfects.

A number is called -hyperperfect if(1)(2)where is the divisor function and the summation is over the proper divisors with . Rearranging gives(3)Taking gives the usual perfect numbers.If is an odd integer, and and are prime, then is -hyperperfect. McCranie (2000) conjectures that all -hyperperfect numbers for odd are in fact of this form. Similarly, if and are distinct odd primes such that for some integer , then is -hyperperfect. Finally, if and is prime, then if is prime for some < then is -hyperperfect (McCranie 2000).The first few hyperperfect numbers (excluding perfect numbers) are 21, 301, 325, 697, 1333, ... (OEIS A007592). If perfect numbers are included, the first few are 6, 21, 28, 301, 325, 496, ... (OEIS A034897), whose corresponding values of are 1, 2, 1, 6, 3, 1, 12, ... (OEIS A034898). The following table gives the first few -hyperperfect numbers for small values of . McCranie (2000) has tabulated all hyperperfect numbers less..

Two numbers are homogeneous if they have identical prime factors. An example of a homogeneous pair is (6, 72), both of which share prime factors 2 and 3:(1)(2)

The term "aliquot divisor" is commonly used to mean two distinct but related things.The first definition is a number that divides another exactly. For instance, 1, 2, 3, and 6 are aliquot divisors of 6. In this sense, "aliquot divisor" is the same as the usual divisor. A number that is not an (aliquot) divisor is said to be an aliquant divisor.The term "aliquot" is also frequently used to specifically mean a proper divisor, i.e., a divisor of a number other than the number itself. For example, the aliquot divisor in this sense of 6 are 1, 2, and 3.

A Smarandache-like function which is defined where is defined as the smallest integer for which . The Smarandache function can therefore be obtained by replacing any factors which are th powers in by their roots.where is the number of solutions to .The functions for , 3, ..., 6 for values such that are tabulated by Begay (1997). The following table gives for small and , 2, ....OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...2A0195541, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ...3A0195551, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ...4A0531661, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, ...

Highly composite numbers are numbers such that divisor function (i.e., the number of divisors of ) is greater than for any smaller . Superabundant numbers are closely related to highly composite numbers, and the first 19 superabundant and highly composite numbers are the same.There are an infinite number of highly composite numbers, and the first few are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, ... (OEIS A002182). The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, ... (OEIS A002183). Ramanujan (1915) listed 102 highly composite numbers up to 6746328388800, but omitted 293318625600. Robin (1983) gives the first 5000 highly composite numbers, and a comprehensive survey is given by Nicholas (1988). Flammenkamp gives a list of the first 779674 highly composite numbers.If(1)is the prime factorization of a highly compositenumber, then 1. The primes 2, 3, ..., form a..

The abundancy of a number is defined as the ratio , where is the divisor function. For , 2, ..., the first few values are 1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, ... (OEIS A017665 and A017666).A positive integer for which is an integer is called a multiperfect number. The first few are 1, 6, 28, 120, 496, 672, 8128, ... (OEIS A007691), corresponding to the abundancies 1, 2, 2, 3, 2, 3, 2, 4, 4, ... (OEIS A054030).

A round number is a number that is the product of a considerable number of comparatively small factors (Hardy 1999, p. 48). Round numbers are very rare. As Hardy (1999, p. 48) notes, "Half the numbers are divisible by 2, one-third by 3, one-sixth by both 2 and 3, and so on. Surely, then we may expect most numbers to have a large number of factors. But the facts seem to show the opposite."A positive integer is sometimes said to be round (or "square root-smooth") if it has no prime factors greater than . The first few such numbers are 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, ... (OEIS A048098). Using this definition, an asymptotic formula for the number of round integers less than or equal to a positive real number is given by(Hildebrand).A different meaning of "round" is used when speaking of "roundinga number."..

The abundance of a number , sometimes also called the abundancy (a term which in this work, is reserved for a different but related quantity), is the quantitywhere is the divisor function. The abundances of , 2, ... are , , , , , 0, , , , , , 4, , , , , ... (OEIS A033880).The following table lists special classifications given to a number based on the value of .classOEISlist of deficient numberA0051001, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, ...almost perfect numberA0000791, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ...0perfect numberA0003966, 28, 496, 8128, ...1quasiperfect numbernone knownabundant numberA00510112, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, ...Values of such that is odd are given by , 2, 4, 8, 9, 16, 18, 25, 32, ... (OEIS A028982; i.e., the union of nonzero squares and twice the squares). Values of such that is square are given by , 12, 28, 70, 88, 108, 168, ... (OEIS A109510)...

An integer whose decimal digits contain no zeros is said to be zerofree. The first few positive zerofree integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ... (OEIS A052382).Zerofree squares are easy to generate, e.g.,(1)Around 1990, D. Hickerson considered the problem of finding large zerofree cubes. After some experimentation, he found a formula that generated infinitely many of them. In March 1998, Bill Gosper asked about 0-free th powers, pointing out that heuristically we should expect there to be infinitely many zerofree squares, cubes, ..., 21st powers, but only finitely many 22nd powers, etc. At this point, Hickerson couldn't locate his formula for cubes, and so came up with the new formula(2)which is 0-free if and .In April 1999, Ed Pegg conjectured on sci.math that there are only finitely many zerofree cubes, so Hickerson posted his new counterexample, (mistakenly claiming that it was the one he had found..

A -repdigit is a number composed of repetition of a single digit (in a given base, generally taken as base 10 unless otherwise specified). For example, the beast number 666 is a (base-10) repdigit. The following table gives the first few repdigits in bases to 10.OEIS-repdigits2A0002251, 3, 7, 15, 31, 63, 127, ...3A0483281, 2, 4, 8, 13, 26, 40, 80, 121, ...4A0483291, 2, 3, 5, 10, 15, 21, 42, 63, 85, 170, ...5A0483301, 2, 3, 4, 6, 12, 18, 24, 31, 62, 93, 124, 156, ...6A0483311, 2, 3, 4, 5, 7, 14, 21, 28, 35, 43, 86, 129, 172, ...7A0483321, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 57, 114, 171, ...8A0483331, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 73, 146, ...9A0483341, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 91, 182, ...10A0107851, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, ...If the digits of a repdigit are all 1s, it is known as a repunit...

To define a recurring digital invariant of order , compute the sum of the th powers of the digits of a number . If this number is equal to the original number , then is called a -Narcissistic number. If not, compute the sums of the th powers of the digits of , and so on. If this process eventually leads back to the original number , the smallest number in the sequence is said to be a -recurring digital invariant. For example,(1)(2)(3)so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy 1979).orderRDIcycle lengths248355, 136, 160, 9193, 2, 3, 241138, 21787, 25244, 8294, 8299, 9044, 9045, 10933,28, 10, 6, 10, 22, 4, 12, 2, 224584, 58618, 89883617148, 63804, 93531, 239459, 28259530, 2, 4, 10, 3780441, 86874, 253074, 376762,92, 56, 27, 30, 14, 21922428, 982108, five more86822, 7973187, 86168049322219, 2274831, 20700388, eleven more1020818070, five more..

Consider the process of taking a number, taking its digit sum, then adding the digits of numbers derived from it, etc., until the remaining number has only one digit. The number of additions required to obtain a single digit from a number in a given base is called the additive persistence of , and the digit obtained is called the digital root of .For example, the sequence obtained from the starting number 9876 in base 10 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a digital root of 3. The base-10 digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888). The digital root of an integer can be computed without actually performing the iteration using the simple congruence formula(1)(2)

A "visual representation" number which is a sum of some simple function of its digits. For example,(1)(2)(3)(4)(5)(6)are all VR numbers given by Madachy (1979), where is a factorial and is a subfactorial.

A rational amicable pair consists of two integers and for which the divisor functions are equal and are of the form(1)where and are bivariate polynomials, and for which the following properties hold (Y. Kohmoto): 1. All the degrees of terms of the numerator of the right fraction are the same. 2. All the degrees of terms of the denominator of the right fraction are the same. 3. The degree of is one greater than the degree of . If and is of the form , then (◇) reduces to the special case(2)so if is an integer, then is a multiperfect number.Consider polynomials of the form(3)For , (◇) reduces to(4)of which no examples are known. For , (◇) reduces to(5)so form an amicable pair. For , (◇) becomes(6)Kohmoto has found three classes of solutions of this type. The first is(7)where is a Mersenne prime with , giving (26403469440047700, 30193441130006700), (7664549986025275200, 8764724625167659200), ... (OEIS A038362 and A038363)...

An -persistent number is a positive integer which contains the digits 0, 1, ..., 9 (i.e., is a pandigital number), and for which , ..., also share this property. No -persistent numbers exist. However, the number is 2-persistent, since but , and the number is 18-persistent. There exists at least one -persistent number for each positive integer .OEIS-persistent1A0512641023456798, 1023456897, 1023456978, 1023456987, ...2A0510181023456789, 1023456879, 1023457689, 1023457869, ...3A0510191052674893, 1052687493, 1052746893, 1052748693, ...4A0510201053274689, 1089467253, 1253094867, 1267085493, ...

Start with an integer , known as the digitaddition generator. Add the sum of the digitaddition generator's digits to obtain the digitaddition . A number can have more than one digitaddition generator. If a number has no digitaddition generator, it is called a self number. The sum of all numbers in a digitaddition series is given by the last term minus the first plus the sum of the digits of the last.If the digitaddition process is performed on to yield its digitaddition , on to yield , etc., a single-digit number, known as the digital root of , is eventually obtained. The digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888).If the process is generalized so that the th (instead of first) powers of the digits of a number are repeatedly added, a periodic sequence of numbers is eventually obtained for any given starting number . For example, the 2-digitaddition sequence for is given by 2, , , , , , , and so on.If..

A number with an even number of digits formed by multiplying a pair of -digit numbers (where the digits are taken from the original number in any order) and together. Pairs of trailing zeros are not allowed. If is a vampire number, then and are called its "fangs." Examples of vampire numbers include(1)(2)(3)(4)(5)(6)(7)(OEIS A014575). The 8-digit vampire numbers are 10025010, 10042510, 10052010, 10052064, 10081260, ... (OEIS A048938) and the 10-digit vampire numbers are 1000174288, 1000191991, 1000198206, 1000250010, ... (OEIS A048939). The numbers of -digit vampires are 0, 7, 148, 3228, ... (OEIS A048935).Vampire numbers having two distinct pairs of fangs include(8)(9)(10)(OEIS A048936).Vampire numbers having three distinct pairs of fangs include(11)(OEIS A048937).The first vampire numbers with four pairs of fangs are(12)(13)(14)(15)and(16)(17)(18)(19)and the first vampire number with five pairs of fangs is(20)(21)(22)(23)(24)(J. K. Andersen,..

A number is said to be pandigital if it contains each of the digits from 0 to 9 (and whose leading digit must be nonzero). However, "zeroless" pandigital quantities contain the digits 1 through 9. Sometimes exclusivity is also required so that each digit is restricted to appear exactly once. For example, 6729/13458 is a (zeroless, restricted) pandigital fraction and 1023456789 is the smallest (zerofull) pandigital number.The first few zerofull restricted pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, ... (OEIS A050278). A 10-digit pandigital number is always divisible by 9 sinceThis passes the divisibility test for 9 since .The smallest unrestricted pandigital primes must therefore have 11 digits (no two of which can be 0). The first few unrestricted pandigital primes are therefore 10123457689, 10123465789, 10123465897, 10123485679, ... (OEIS A050288).If zeros are excluded, the..

Let be the sum of the base- digits of , and the Thue-Morse sequence, then

A number of the form , , etc. The first few nontrivial undulants (with the stipulation that ) are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, ... (OEIS A046075). Including the trivial 1- and 2-digit undulants and dropping the requirement that gives OEIS A033619.The first few undulating squares are 121, 484, 676, 69696, ... (OEIS A016073), with no larger such numbers of fewer than a million digits (Pickover 1995). Several tricks can be used to speed the search for square undulating numbers, especially by examining the possible patterns of ending digits. For example, the only possible sets of four trailing digits for undulating squares are 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and 9696.The only undulating power for and up to 100 digits is (Pickover 1995). A large undulating prime is given by (Pickover 1995).A binary undulant is a power of 2 whose base-10 representation contains one or both of the sequences and . The first few..

A number such that the last digits of are the same as . 49 is trimorphic since (Wells 1986, p. 124). The first few are 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, ... (OEIS A033819).

Let be the number of digit blocks of a sequence in the base- expansion of . The following table gives the sequence for a number of blocks .OEIS for , 2, ...00A0569730, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, ...01A0378000, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, ...10A0332640, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, ...11A0140810, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 0, ...000A0569740, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, ...001A0569750, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...010A0569760, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ...011A0569770, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...100A0569780, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, ...101A0569790, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, ...110A0569800, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, ...111A0140820, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, .....

A tetradic (or four-way) number is a number that remains unchanged when flipped back to front, mirrored up-down, or flipped up-down. Since the only numbers that remain unchanged which turned up-side-down or mirrored are 0, 1, and 8 (here, the numerals 1 and 8 are assumed to be written as a single stroke and symmetrical pair of loops, respectively), a tetradic number is precisely a palindromic number containing only 0, 1, and 8 as digits. The first few are therefore 1, 8, 11, 88, 101, 111, 181, 808, 818, ... (OEIS A006072).The first few tetradic primes are 11, 101, 181, 18181, 1008001, 1180811, 1880881, 1881881, ... (OEIS A068188). The largest known tetradic prime as of Apr. 2010 iswhere is a repunit (https://primes.utm.edu/top20/page.php?id=53#records), which has decimal digits.

An -digit number that is the sum of the th powers of its digits is called an -narcissistic number. It is also sometimes known as an Armstrong number, perfect digital invariant (Madachy 1979), or plus perfect number. Hardy (1993) wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits: , , , and . These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician." Narcissistic numbers therefore generalize these "unappealing" numbers to other powers (Madachy 1979, p. 164).The smallest example of a narcissistic number other than the trivial 1-digitnumbers is(1)The first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208,9474, 54748, ... (OEIS A005188).It can easily be shown that base-10 -narcissistic numbers can exist only for , since(2)for . In fact, as summarized in the table..

A Münchhausen number (sometimes spelled Münchausen number, with a single 'h') is a number equal to the sum of its digits raised to each digit's power. Münchhausen numbers therefore differ from narcissistic numbers, which are numbers that equal the sum of a fixed power (in particular, the number of decimal digits) of the given number. The name "Münchhausen number" derives from the fact that these numbers "raise themselves" analogously to way in which Baron Hieronymus von Münchhausen allegedly raised himself by riding a cannonball, as portrayed in the 1943 fantasy comedy film Münchhausen.If 0s are disallowed (since is not well-defined), the only Münchhausen numbers are 1 andIf the definition is adopted, then there are exactly four Münchhausen numbers: 0, 1, 3435, and 438579088 (OEIS A046253)...

Numbers which are not perfect and for whichor equivalentlywhere is the divisor function. Deficient numbers are sometimes called defective numbers (Singh 1997). Primes, prime powers, and any divisors of a perfect or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, ... (OEIS A005100).

Consider the process of taking a number, multiplying its digits, then multiplying the digits of numbers derived from it, etc., until the remaining number has only one digit. The number of multiplications required to obtain a single digit from a number is called the multiplicative persistence of , and the digit obtained is called the multiplicative digital root of .For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has a multiplicative persistence of two and a multiplicative digital root of 0. The multiplicative digital roots of the first few positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, ... (OEIS A031347).OEISnumbers having multiplicative digital root 0A0340480, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, ...1A0022751, 11, 111, 1111, 11111, 111111, 1111111, 11111111, ...2A0340492, 12, 21, 26, 34, 37, 43, 62, 73, 112, 121, 126, ...3A0340503,..

A metadrome is a number whose hexadecimal digits are in strict ascending order. The first few are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, ... (OEIS A023784).A number that is not a metadrome is a nialpdrome.The following table summarized related classes of numbers.namebase-16 digit orderkatadromestrict descendingmetadromestrict ascendingnialpdromenonincreasingplaindromenondecreasing

A Smarandache prime is a prime Smarandache number, i.e., a prime number of the form . Surprisingly, no Smarandache primes are known as of Nov. 2015. Upper limits on the non-appearance of primes are summarized in the table below. The search from to was completed by Balatov (2015b), and search of larger terms is now underway (Great Smarandache PRPrime search). As of Dec. 2016, it is known that there are no Smarandache primes up to index 344869.reference200Fleuren (1999)E. Weisstein (Mar. 21, 2009)E. Weisstein (Oct. 17, 2011)M. Alekseyev (Oct. 3, 2015)S. Batalov (Oct. 22 2015)The Great Smarandache PRPrime search (Dec. 5, 2016)If all digit substrings are allowed (so that e.g., concatenating just the 1 from 10, just 10111 from 101112, etc. are permitted), prime digit sequences are known. In particular, such primes are Champernowne-constant primes, the first few of which..

A number such that has its last digit(s) equal to is called -automorphic. For example, (Wells 1986, pp. 58-59) and (Wells 1986, p. 68), so 5 and 6 are 1-automorphic. Similarly, and , so 8 and 88 are 2-automorphic. de Guerre and Fairbairn (1968) give a history of automorphic numbers.The first few 1-automorphic numbers are 1, 5, 6, 25, 76, 376, 625, 9376, 90625, ... (OEIS A003226, Wells 1986, p. 130). There are two 1-automorphic numbers with a given number of digits, one ending in 5 and one in 6 (except that the 1-digit automorphic numbers include 1), and each of these contains the previous number with a digit prepended. Using this fact, it is possible to construct automorphic numbers having more than digits (Madachy 1979). The first few 1-automorphic numbers ending with 5 are 5, 25, 625, 0625, 90625, ... (OEIS A007185), and the first few ending with 6 are 6, 76, 376, 9376, 09376, ... (OEIS A016090). The 1-automorphic numbers ending in..

Consider the consecutive number sequences formed by the concatenation of the first positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to are given by(1)(2)These are sometimes called Smarandache consecutive numbers, but in this work, the terms in the sequence will be called simply Smarandache numbers. Similarly, a Smarandache number that is prime will be called a Smarandache prime. Surprisingly, no Smarandache primes exist for (Great Smarandache PRPrime search; Dec. 5, 2016).The number of digits of can be computed by noticing the pattern in the following table, where(3)is the number of digits in . rangedigits11-9210-993100-99941000-9999By..

A pair of numbers and such thatwhere is the divisor function. Beck and Najar (1977) found 11 augmented amicable pairs.

A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation(1)for , 3, ..., where . The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052).An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence(2)for , 2, ..., where and is the number of digits in . An infinite number of -self numbers can be generated from the sequence(3)for , 3, ..., and(4)Joshi (1973) proved that if is odd, then is a -self number iff is odd. Patel (1991) proved that , , and are -self numbers in every even base ...

A katadrome is a number whose hexadecimal digits are in strict descending order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 32, 33, 48, 49, ... (OEIS A023797), corresponding to 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 20, 21, 30, 31, ....A number that is not a katadrome is a plaindrome.The following table summarized related classes of numbers.namebase-16 digit orderkatadromestrict descendingmetadromestrict ascendingnialpdromenonincreasingplaindromenondecreasing

The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. Now compute (discarding any initial 0s) and iterate, where is sometimes called the Kaprekar function. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending on the number of digits in and the value of . The list of values is sometimes called a Kaprekar sequence, and the result is sometimes called a Kaprekar number (Deutsch and Goldman 2004), though this nomenclature should be deprecated because of confusing with the distinct sort of Kaprekar number.In base-10, the numbers for which are given by 495, 6174, 549945, 631764, ... (OEIS A099009). Similarly, the numbers for which iterating gives a cycle of length are given by 53955, 59994, 61974, 62964, 63954, 71973, ... (OEIS..

Consider an -digit number . Square it and add the right digits to the left or digits. If the resultant sum is , then is called a Kaprekar number. For example, 9 is a Kaprekar number sinceand 297 is a Kaprekar number sinceThe first few are 1, 9, 45, 55, 99, 297, 703, ... (OEIS A006886).

Dickson (1913, 2005) defined an amicable triple to be a triple of three numbers such that(1)(2)(3)where is the restricted divisor function (Madachy 1979). Dickson (1913, 2005) found eight sets of amicable triples with two equal numbers, and two sets with distinct numbers. The latter are (123228768, 103340640, 124015008), for which(4)(5)(6)and (1945330728960, 2324196638720, 2615631953920), for which(7)(8)(9)(10)(11)(12)A second definition (Guy 1994) defines an amicable triple as a triple such that(13)where is the divisor function. An example is (, , ).

A number is called amenable if it can be built up from integers , , ..., by either addition or multiplication such that(1)(Tamvakis 1995).The solutions are the numbers such that or 1 (mod 4), excluding (Lossers 1998), giving 1, 5, 8, 9, 12, 13, 16, 17, ... (OEIS A100832). For example, 5 and 8 are amenable since(2)(3)(4)(5)

A positive integer is called a base- Rhonda number if the product of the base- digits of is equal to times the sum of 's prime factors. These numbers were named by K. S. Brown after an acquaintance of his whose residence number 25662 satisfies this property. The etymology of the term is therefore similar to the Smith numbers.25662 is a Rhonda number to base-10 since its prime factorization is(1)and the product of its base-10 digits satisfies(2)The Rhonda numbers to base 10 are 1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, ... (OEIS A099542). The corresponding sums of prime factors are 24, 24, 28, 30, 54, 72, 32, 24, 48, 72, ... (OEIS A099543).Rhonda numbers exist only for bases that are composite since there is no way for the product of integers less than a prime to have as a factor.The first few Rhonda numbers for small composite bases are summarized in the following table.OEIS-Rhonda numbers4A10096810206, 11935, 12150,..

A composite number defined analogously to a Smith number except that the sum of the number's digits equals the sum of the digits of its distinct prime factors (excluding 1).The first few hoax numbers are 22, 58, 84, 85, 94, 136, 160, 166, 202, 234, ... (OEIS A019506), illustrated above as a binary plot, and the corresponding sums of digits are 4, 13, 12, 13, 13, 10, 7, 13, 4, 9, 7, ... (OEIS A050223).

A positive integer which is divisible by the sum of its digits, also called a Niven number (Kennedy et al. 1980) or a multidigital number (Kaprekar 1955). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (OEIS A005349). Grundman (1994) proved that there is no sequence of more than 20 consecutive Harshad numbers, and found the smallest sequence of 20 consecutive Harshad numbers, each member of which has digits.Grundman (1994) defined an -Harshad (or -Niven) number to be a positive integer which is divisible by the sum of its digits in base . Cai (1996) showed that for or 3, there exists an infinite family of sequences of consecutive -Harshad numbers of length .Define an all-Harshad (or all-Niven) number as a positive integer which is divisible by the sum of its digits in all bases . Then only 1, 2, 4, and 6 are all-Harshad numbers...

Let the sum of the squares of the digits of a positive integer be represented by . In a similar way, let the sum of the squares of the digits of be represented by , and so on.Iterating this sum-of-squared-digits map always eventually reaches one of the 10 numbers 0, 1, 4, 16, 20, 37, 42, 58, 89, or 145 (OEIS A039943; Porges 1945).If for some , then the original integer is said to be happy. For example, starting with 7 gives the sequence 7, 49, 97, 130, 10, 1, so 7 is a happy number.The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, ... (OEIS A007770). These are also the numbers whose 2-recurring digital invariant sequences have period 1. The numbers of iterations required for these to reach 1 are 0, 5, 1, 2, 4, 3, 3, 2, 3, 4, 4, 2, 5, ... (OEIS A090425).The numbers of happy numbers less than or equal to 1, , , ... are given by 1, 3, 20, 143, 1442, 14377, 143071, ... (OEIS A068571).The first few consecutive happy numbers have..

Consider the process of taking a number, adding its digits, then adding the digits of the number derived from it, etc., until the remaining number has only one digit. The number of additions required to obtain a single digit from a number is called the additive persistence of , and the digit obtained is called the digital root of .For example, the sequence obtained from the starting number 9876 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a digital root of 3. The additive persistences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, ... (OEIS A031286). The smallest numbers of additive persistence for , 1, ... are 0, 10, 19, 199, 19999999999999999999999, ... (OEIS A006050).

The reversal of a positive integer is . The reversal of a positive integer is implemented in the Wolfram Language as IntegerReverse[n].A positive integer that is the same as its own reversal is known as a palindromicnumber.Ball and Coxeter (1987) consider numbers whose reversals are integral multiples of themselves. Palindromic numbers and numbers ending with a zero are trivial examples.The first few nontrivial examples of numbers whose reversals are multiples of themselves are 8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, ... (OEIS A031877). The pattern continues for large numbers, with numbers of the form equal to 4 times their reversals and numbers of the form equal to 9 times their reversals. In addition, runs of numbers of either of these forms can be concatenated to yield numbers of the form , equal to 4 times their reversals, and , equal to 9 times their reversals.The reversals..

An abundant number, sometimes also called an excessive number, is a positive integer for which(1)where is the divisor function and is the restricted divisor function. The quantity is sometimes called the abundance.A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).Every positive integer with is abundant. Any multiple of a perfect number or an abundant number is also abundant. Prime numbers are not abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.There are only 21 abundant numbers less than 100, and they are all even.The first odd abundant number is(2)That 945 is abundant can be seen by computing(3)Define the density function(4)(correcting the expression in Finch 2003, p. 126) for a positive real number where gives the cardinal..

The quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."

1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):(K. MacMillan, pers. comm., Apr. 29, 2007).This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.1729 was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character..

The first strong law of small numbers (Gardner 1980, Guy 1988, 1990) states "There aren't enough small numbers to meet the many demands made of them."The second strong law of small numbers (Guy 1990) states that "When two numbers look equal, it ain't necessarily so." Guy (1988) gives 35 examples of this statement, and 40 more in Guy (1990). For example, example 35 notes that the first few values of the interpolating polynomial (erroneously given in Guy (1990) with a coefficient 24 instead of 23) for , 2, ... are 1, 2, 4, 8, 16, .... Thus, the polynomial appears to give the powers of 2, but then continues 31, 57, 99, ... (OEIS A000127). In fact, this sequence gives the maximal number of regions obtained by joining points around a circle by chords (circle division by chords).Similarly, example 41 notes the curious fact that the function where is the ceiling function gives the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (i.e., the first..

Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Schroeppel (1972). 239 appears in Machin's formula(1)which is related to the fact that(2)which is why 239/169 is the 7th convergent of . Another pair of inverse tangent formulas involving 239 is(3)(4)239 needs 4 squares (the maximum) to express it, 9 cubes (the maximum, shared only with 23) to express it, and 19 fourth powers (the maximum) to express it (see Waring's problem). However, 239 doesn't need the maximum number of fifth powers (Beeler et al. 1972, Item 63).

Guy's "strong law of small numbers" states that there aren't enough small numbers to meet the many demands made of them. Guy (1988) also gives several interesting and misleading facts about small numbers: 1. 10% of the first 100 numbers are square numbers.2. A quarter of the numbers are primes. 3. All numbers less than 10, except for 6, are prime powers. 4. Half the numbers less than 10 are Fibonacci numbers.

The number 163 is very important in number theory, since is the largest number such that the imaginary quadratic field has class number . It also satisfies the curious identities(1)(2)(3)where is a binomial coefficient.

The second Mersenne prime , which is itself the exponent of Mersenne prime . It gives rise to the perfect number It is a Gaussian prime, but not an Eisenstein prime, since it factors as , where is a primitive cube root of unity. It is the smallest non-Sophie Germain prime. It is also the smallest non-Fermat prime, and as such is the smallest number of faces of a regular polygon (the heptagon) that is not constructible by straightedge and compass.It occurs as a sacred number in the Bible and in various other traditions. In Babylonian numerology it was considered as the perfect number, the only number between 2 and 10 which is not generated (divisible) by any other number, nor does it generate (divide) any other number.Words referring to number seven may have the prefix hepta-, derived from the Greek -) (heptic), or sept- (septuple), derived from the Latin septem...

The smallest composite squarefree number (), and the third triangular number (). It is the also smallest perfect number, since . The number 6 arises in combinatorics as the binomial coefficient , which appears in Pascal's triangle and counts the 2-subsets of a set with 4 elements. It is also equal to (3 factorial), the number of permutations of three objects, and the order of the symmetric group (which is the smallest non-Abelian group).Six is indicated by the Latin prefix sex-, as in sextic, or by the Greek prefix hexa- (-), as in hexagon, hexagram, or hexahedron.The six-fold symmetry is typical of crystals such as snowflakes. A mathematical and physical treatment can be found in Kepler (Halleux 1975), Descartes (1637), Weyl (1952), and Chandrasekharan (1986).

According to the novel The Hitchhiker's Guide to the Galaxy (Adams 1997), 42 is the ultimate answer to life, the universe, and everything. Unfortunately, it is left as an exercise to the reader to determine the actual question.On Feb. 18, 2005, the 42nd Mersenne prime was discovered (Weisstein 2005), leading to humorous speculation that the answer to life, the universe, and everything is somehow contained in the 7.8 million decimal digits of that number.It is also amusing that 042 occurs as the digit string ending at the 50 billionth decimal place in each of and , providing another excellent answer to the ultimate question (Berggren et al. 1997, p. 729).

The third prime number, which is also the second Fermat prime, the third Sophie Germain prime, and Fibonacci number . It is an Eisenstein prime, but not a Gaussian prime, since it factors as . It is the hypotenuse of the smallest Pythagorean triple: 3, 4, 5. For the Pythagorean school, the number 5 was the number of marriage, since it is was the sum of the first female number (2) and the first male number (3). The magic symbol of the pentagram was also based on number 5; it is a star polygon with the smallest possible number of sides, and is formed by the diagonals of a regular pentagon. These intersect each other according to the golden ratio .There are five Platonic solids. In algebra, five arises in Abel's impossibility theorem as the smallest degree for which an algebraic equation with general coefficients is not solvable by radicals. According to Galois theory, this property is a consequence of the fact that 5 is the smallest positive integer such that..

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of the Pythagorean school, it was the number of justice. The sacred tetraktýs (10) was the sum of the first four numbers, depicted as a triangle with two equal sides of length 4.4 is the highest degree for which an algebraic equation is always solvable by radicals. It is the smallest order of a field which is not a prime field, and the smallest order for which there exist two nonisomorphic finite groups (finite group C2×C2 and the cyclic group C4). It is the smallest number of faces of a regular polyhedron, the tetrahedron. In the three-dimensional Euclidean space, there is exactly one sphere passing through four noncoplanar points. Four is the number of dimensions of space-time.Words related to number four are indicated by the Greek prefix tetra (e.g., tetromino) or by..

3 is the only integer which is the sum of the preceding positive integers () and the only number which is the sum of the factorials of the preceding positive integers (). It is also the first odd prime. A quantity taken to the power 3 is said to be cubed.The sequence 1, 31, 331, 3331, 33331, ... (OEIS A033175) consisting of , 1, ... 3s followed by a 1 has its th term is given byThe result is prime for , 2, 3, 4, 5, 6, 7, 17, 39, ... (OEIS A055520); i.e., for 31, 331, 3331, 33331, 333331, 3333331, 33333331, ... (OEIS A051200), a fact which Gardner (1997) calls "a remarkable pattern that is entirely accidental and leads nowhere."

The number one (1), also called "unity," is the first positive integer. It is an odd number. Although the number 1 used to be considered a prime number, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own (Wells 1986, p. 31). The number 1 is sometimes also called "unity," so the th roots of 1 are often called the th roots of unity. Fractions having 1 as a numerator are called unit fractions. If only one root, solution, etc., exists to a given problem, the solution is called unique.The generating function having all coefficients1 is given by(1)The number one is also equivalent to the repeatingdecimal(2)(3)

A plaindrome is a number whose hexadecimal digits are in nondecreasing order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, ... (OEIS A023757).A number that is not a plaindrome is called a katadrome.The following table summarizes related classes of numbers.namebase-16 digit orderkatadromestrict descendingmetadromestrict ascendingnialpdromenonincreasingplaindromenondecreasing

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number with decimal expansion 1234.56 is defined as(1)(2)Expressions written in this form (where negative are allowed as exemplified above but usually not considered in elementary education contexts) are said to be in expanded notation.Other examples include the decimal expansion of given by 625, of given by 3.14159..., and of given by 0.1111.... The decimal expansion of a number can be found in the Wolfram Language using the command RealDigits[n], or equivalently, RealDigits[n, 10].The decimal expansion of a number may terminate (in which case the number is called a regular number or finite decimal, e.g., ), eventually..

The base 8 notational system for representing real numbers. The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that (8 in base 10) is represented as () in base 8. The following table gives the octal equivalents of the first few decimal numbers.11111321252212142226331315232744141624305515172531661620263277172127338101822283491119232935101220243036The song "New Math" by Tom Lehrer (That Was the Year That Was, 1965) explains how to compute in octal. (The answer is .)

The length of a number in base is the number of digits in the base- numeral for , given by the formulawhere is the floor function.The multiplicative persistence of an -digit is sometimes also called its length.

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