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Fermat's 4n+1 theorem

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...

Pandigital fraction

A fraction containing each of the digits 1 through 9 is called a pandigital fraction. The following table gives the number of pandigital fractions which represent simple unit fractions. The numbers of pandigital fractions for 1/1, 1/2, 1/3, ... are 0, 12, 2, 4, 12, 3, 7, 46, 3, ... (OEIS A054383).#fractions12,2412,37,46,,,,,,,3004

Egyptian number

A number is called an Egyptian number if it is the sum of the denominators in some unit fraction representation of a positive whole number not consisting entirely of 1s. For example,so is an Egyptian number. The numbers that are not Egyptian are 2, 3, 5, 6, 7, 8, 12, 13, 14, 15, 19, 21, and 23 (OEIS A028229; Konhauser et al. 1996, p. 147).If is the sum of denominators of a unit fraction representation composed of distinct denominators which are not all 1s, then it is called a strictly Egyptian number. For example, by virtue of is Egyptian, but it is not strictly Egyptian. Graham (1963) proved that every number is strictly Egyptian. Numbers that are strictly Egyptian are 11, 24, 30, 31, 32, 37, 38, 43, ... (OEIS A052428), and those which are not are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ... (OEIS A051882)...

Pascal matrix

Three types of matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix with , a lower triangular matrix with , and an upper triangular matrix with , where , 1, ..., . For example, for , these would be given by(1)(2)(3)The Pascal -matrix or order is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].These matrices have some amazing properties. In particular, their determinants are all equal to 1(4)and(5)(Edelman and Strang).Edelman and Strang give four proofs of the identity (5), themost straightforward of which is(6)(7)(8)(9)where Einstein summation has been used.

Trinomial triangle

The trinomial triangle is a number triangle of trinomial coefficients. It can be obtained by starting with a row containing a single "1" and the next row containing three 1s and then letting subsequent row elements be computed by summing the elements above to the left, directly above, and above to the right:(OEIS A027907).The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened trinomial triangle.

Sárkőzy's theorem

A partial solution to the Erdős squarefree conjecture which states that the binomial coefficient is never squarefree for all sufficiently large . Sárkőzy (1985) showed that if is the square part of the binomial coefficient , thenwhere is the Riemann zeta function. An upper bound on of has been obtained.

Plato's numbers

The positive integers 216 and appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to (Livio 2002, p. 66).In Laws, Plato suggests that is the optimal number of citizens in a state because 1. It is the product of 12, 20, and 21. 2. The 12th part of it can still be divided by 12. 3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11 (Livio 2002, p. 65).

Rabbit constant

The limiting rabbit sequence written as a binary fraction (OEIS A005614), where denotes a binary number (a number in base-2). The decimal value is(1)(OEIS A014565).Amazingly, the rabbit constant is also given by the continued fraction [0; , , , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where are Fibonacci numbers with taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence by(2)where is the floor function and is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then(3)This is a special case of the Devil's staircase function with .The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).

Eulerian number

The Eulerian number gives the number of permutations of having permutation ascents (Graham et al. 1994, p. 267). Note that a slightly different definition of Eulerian number is used by Comtet (1974), who defines the Eulerian number (sometimes also denoted ) as the number of permutation runs of length , and hence .The Eulerian numbers are given explicitly by the sum(1)(Comtet 1974, p.  243). The Eulerian numbers satisfy the sum identity(2)as well as Worpitzky's identity(3)Eulerian numbers also arise in the surprising context of integrating the sincfunction, and also in sums of the form(4)(5)where is the polylogarithm function. is therefore given by the coefficient of in(6) has the exponential generating function(7)The Eulerian numbers satisfy the recurrence relation(8)Special cases are given by(9)(10)(11)and summarized in the following table.OEIS, , , ...1A0002950, 1, 4, 11, 26, 57, 120, 247, 502, 1013, ...2A0004600,..

Langford's problem

Arrange copies of the digits 1, ..., such that there is one digit between the 1s, two digits between the 2s, etc. For example, the unique (modulo reversal) solution is 231213, and the unique (again modulo reversal) solution is 23421314. Solutions to Langford's problem exist only if , so the next solutions occur for . There are 26 of these, as exhibited by Lloyd (1971). In lexicographically smallest order (i.e., small digits come first), the first few Langford sequences are 231213, 23421314, 14156742352637, 14167345236275, 15146735423627, ... (OEIS A050998).The number of solutions for , 4, 5, ... (modulo reversal of the digits) are 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, ... (OEIS A014552). No formula is known for the number of solutions of a given order .

Rabbit sequence

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map correspond to young rabbits growing old, and correspond to old rabbits producing young rabbits. Starting with 0 and iterating using string rewriting gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... A recurrence plot of the limiting value of this sequence is illustrated above.Converted to decimal, this sequence gives 1, 2, 5, 22, 181, ... (OEIS A005203), with the th term given by the recurrence relationwith , , and the th Fibonacci number.The limiting sequence written as a binary fraction (OEIS A005614), where denotes a binary number (i.e., a number written in base 2, so or 1), is called the rabbit constant.

Least significant bit

The value of the bit in a binary number. For the sequence of numbers 1, 2, 3, 4, ..., the least significant bits are therefore the alternating sequence 1, 0, 1, 0, 1, 0, ... (OEIS A000035). It can be represented as(1)(2)or(3)It is also given by the linear recurrenceequation(4)with (Wolfram 2002, p. 128).Analogously, the "most significant bit" is the value of the bit in an -bit representation.The least significant bit has Lambert series(5)where is a q-polygamma function.

Bit length

The number of binary bits necessary to represent a number, given explicitly by(1)(2)where is the ceiling function, is the floor function, and is lg, the logarithm to base 2. For , 2, ..., the sequence of bit lengths is given by 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, ... (OEIS A036377). The function is given by the Wolfram Language function BitLength[n].

Large number

A wide variety of large numbers crop up in mathematics. Some are contrived, but some actually arise in proofs. Often, it is possible to prove existence theorems by deriving some potentially huge upper limit which is frequently greatly reduced in subsequent versions (e.g., Graham's number, Kolmogorov-Arnold-Moser theorem, Mertens conjecture, Skewes number, Wang's conjecture).Large decimal numbers beginning with are named according to two mutually conflicting nomenclatures: the American system (in which the prefix stands for in ) and the British system (in which the prefix stands for in ). The British names for billion, trillion, etc. originate from the late 15th century when the French physician and mathematician Nicolas Chuquet (1445-1488) used the Latin prefixes to denote successive powers of one million () and the suffix "-llion" to refer one million (Rowlett). In more recent years, the "American" system..

Number

The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a complex number, integer, real number, etc.?). Wherever possible in this work, the word "number" is used to refer to quantities which are integers, and "constant" is reserved for nonintegral numbers which have a fixed value. Because terms such as real number, Bernoulli number, and irrational number are commonly used to refer to nonintegral quantities, however, it is not possible to be entirely consistent in nomenclature.To indicate a particular numerical label, the abbreviation "no." is sometimes used (deriving from "numero," the ablative case of the Latin "numerus"), as is the less common "nr." The symbol # (known as the octothorpe) is commonly used to denote "number."While..

Semiprime

A semiprime, also called a 2-almost prime, biprime (Conway et al. 2008), or -number, is a composite number that is the product of two (possibly equal) primes. The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (OEIS A001358). The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, 10, 14, 15, 21, 22, 26, 33, 34, ... (OEIS A006881).The square of any prime number is by definition a semiprime. The largest known semiprime is therefore the square of the largest known prime.A formula for the number of semiprimes less than or equal to is given by(1)where is the prime counting function and is the th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006).The numbers of semiprimes less than for , 2, ... are 3, 34, 299, 2625, 23378, 210035, ... (OEIS A066265).For with and distinct, the following..

Sierpiński constant

Let the sum of squares function denote the number of representations of by squares, then the summatory function of has the asymptotic expansion(1)where(2)(3)(4)(5)(OEIS A241017) is the Sierpiński constant (Finch 2003, p. 123), is the Dirichlet beta function, is the Euler-Mascheroni constant, and is the gamma function.

Hilbert number

The Gelfond-Schneider constant is sometimesknown as the Hilbert number.Flannery and Flannery (2000, p. 35) define a Hilbert number as a positive integer of the form (i.e., a positive integer such that ). The first few are then 1, 5, 9, 13, 17, 21, 25, ... (OEIS A016813). A Hilbert number that is not divisible by a smaller Hilbert number (other than 1) is then called a Hilbert prime (or S-prime; Apostol 1976, p. 101); otherwise, is called a Hilbert composite. The first few Hilbert primes are 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (OEIS A057948), and the first few Hilbert composites are 25, 45, 65, 81, 85, ... (OEIS A054520).Factorization with respect to Hilbert primes is not necessarily unique, as illustrated by the exampleThe first few such examples are 441, 693, 1089, 1197, 1449, ... (OEIS A057949)...

Fibonacci factorial constant

The Fibonacci factorial constant is the constant appearing in the asymptotic growth of the fibonorials (aka. Fibonacci factorials) . It is given by the infinite product(1)where(2)and is the golden ratio.It can be given in closed form by(3)(4)(5)(OEIS A062073), where is a q-Pochhammer symbol and is a Jacobi theta function.

Square

The term "square" can be used to mean either a square number (" is the square of ") or a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other as illustrated above. In other words, a square is a regular polygon with four sides.When used as a symbol, denotes a square geometric figure with given vertices, while is sometimes used to denote a graph product (Clark and Suen 2000).A square is a special case of an isosceles trapezoid, kite, parallelogram, quadrilateral, rectangle, rhombus, and trapezoid.The diagonals of a square bisect one another and are perpendicular (illustrated in red in the figure above). In addition, they bisect each pair of opposite angles (illustrated in blue).The perimeter of a square with side length is(1)and the area is(2)The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon..

Jevons' number

A semiprime which English economist and logician William Stanley Jevons incorrectly believed no one else would be able to factor. According to Jevons (1874, p. 123), "Can the reader say what two numbers multiplied together will produce the number 8616460799? I think it unlikely that anyone but myself will ever know."Actually, a modern computer can factor this number in a few milliseconds as the product of two five-digit numbers:Published factorizations include those by Lehmer (1903) and Golomb (1996).

Narayana number

The Narayan number for , 2, ... and , ..., gives a solution to several counting problems in combinatorics. For example, gives the number of expressions with pairs of parentheses that are correctly matched and contain distinct nestings. It also gives the number Dyck paths of length with exactly peaks.A closed-form expression of is given bywhere is a binomial coefficient.Summing over gives the Catalan numberEnumerating as a number triangle is called the Narayana triangle.

Catalan's triangle

Catalan's triangle is the number triangle(1)(OEIS A009766) with entries given by(2)for . Each element is equal to the one above plus the one to the left. The sum of each row is equal to the last element of the next row and also equal to the Catalan number . Furthermore, .The coefficients also give the number of nonnegative partial sums of 1s and s, denoted by Bailey (1996), who gave the alternate form(3)(4)for .

Evil number

A number in which the first decimal digits of the fractional part sum to 666 is known as an evil number (Pegg and Lomont 2004).However, the term "evil" is also used to denote nonnegative integers that have an even number of 1s in their binary expansions, the first few of which are 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, ... (OEIS A001969), illustrated above as a binary plot. Numbers that are not evil are then known as odious numbers.Returning to Pegg's definition of evil, the fact that is evil was noted by Keith, while I. Honig (pers. comm., May 9, 2004) noted that the golden ratio is also evil. The following table gives a list of some common evil numbers (Pegg and Lomont 2004).Ramanujan constant 132hard hexagon entropy constant 137139140Stieltjes constant 142pi 144golden ratio 146146151Glaisher-Kinkelin constant 153cube line picking average length155Delian constant 156The probability of the digits of a given real number summing..

Strong pseudoprime

A strong pseudoprime to a base is an odd composite number with (for odd) for which either(1)or(2)for some , 1, ..., (Riesel 1994, p. 91). Note that Guy (1994, p. 27) restricts the definition of strong pseudoprimes to only those satisfying (1).The definition is motivated by the fact that a Fermat pseudoprime to the base satisfies(3)But since is odd, it can be written , and(4)If is prime, it must divide at least one of the factors, but can't divide both because it would then divide their difference(5)Therefore,(6)so write to obtain(7)If divides exactly one of these factors but is composite, it is a strong pseudoprime. A composite number is a strong pseudoprime to at most 1/4 of all bases less than itself (Monier 1980, Rabin 1980). The strong pseudoprimes provide the basis for Miller's primality test and Rabin-Miller strong pseudoprime test.A strong pseudoprime to the base is also an Euler pseudoprime to the base (Pomerance et al. 1980)...

Constant primes

Let be a prime with digits and let be a constant. Call an "-prime" if the concatenation of the first digits of (ignoring the decimal point if one is present) give . Constant primes are therefore a special type of integer sequence primes, with e-primes, pi-primes, and phi-primes being perhaps the most prominent examples.The following table summarizes the indices of known constant primes for some named mathematical constants.constantname of primesOEIS giving primeApéry's constantA11933410, 55, 109, 141Catalan's constantA11832852, 276, 25477Champernowne constantA07162010, 14, 24, 235, 2804, 4347, 37735, 68433Copeland-Erdős constantA2275301, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048ee-primeA0641181, 3, 7, 85, 1781, 2780, 112280, 155025Euler-Mascheroni constantA0658151, 3, 40, 185, 1038, 22610, 179849Glaisher-Kinkelin constantA1184207, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692Golomb-Dickman..

Wallis's problem

Find nontrivial solutions to other than , where is the divisor function. Nontrivial solutions means that solutions which are multiples of smaller solutions are not considered. For example, multiples of are solutions for , 7, 9, 11, 13, 17, 19, 23, 21, ....Nontrivial solutions to Wallis's equation include , (326, 407), (406, 489), (627, 749), (740, 878), (880, 1451), (888, 1102), (1026, 1208), (1110, 1943), (1284, 1528, 1605), (1510, 1809), (1628, 1630, 2035), (1956, 2030, 2445), (2013, 2557), (2072, 3097), (2508, 2996, 3135, 3745), ....

Pillai's conjecture

For every , there exist only finite many pairs of powers with and natural numbers and .

Factorion

A factorion is an integer which is equal to the sum of factorials of its digits. There are exactly four such numbers:(1)(2)(3)(4)(OEIS A014080; Gardner 1978, Madachy 1979, Pickover 1995). Obviously, the factorion of an -digit number cannot exceed .

Legion's numbers

Legion's number of the first kind is defined as(1)(2)where 666 is the beast number. It has 1881 decimaldigits.Legion's number of the second kind is defined as(3)(4)It has approximately digits, and ends with trailing zeros.

Primefree sequence

A primefree sequence is sequence whose terms are never prime. Graham (1964) proved that there exist relatively prime positive integers and such that the recurrence equation(1)with and contains no prime numbers.In addition, Graham (1964) constructed a pair of numbers (one 33 digits and the other 34)(2)(3)satisfying this condition. Knuth (1990) subsequently found a 17-digit pair(4)(5)satisfying the same conditions. Almost immediately, Wilf (1990) found a smaller pair (one 17 digits and the other 16)(6)(7)Note that Hoffman (1998, p. 159) inadvertently inverted the order of the Wilf (1990) pair, thus obtaining a sequence that has prime terms for , 163, 190, 523, 1855, 3228, 3579, 6468, 7170, 10230, 12783, 17259, 60139, 91315, 97923, 101823, 156075, 182220, ... (OEIS A108156), with no others for (E. W. Weisstein, May 5, 2006).Nicol (1999) subsequently found the 12-digit pair ...

Power floor prime sequence

A power floor prime sequence is a sequence of prime numbers , where is the floor function and is real number. It is unknown if, though extremely unlikely that, infinite sequences of this type exist. An example having eight consecutive primes is , which gives 2, 5, 13, 31, 73, 173, 409, and 967 and has the smallest possible numerator and denominators for an 8-term sequence (D. Terr, pers. comm., Sep. 1, 2004). D. Terr (pers. comm., Jan. 21, 2003) has found a sequence of length 100.

Digit

The number of digits in an integer is the number of numbers in some base (usually 10) required to represent it. The numbers 1 to 9 are therefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit inflation" are occasionally encountered, although this particular usage has thankfully not been needed in the U.S. for some time. The number of base- digits in a number can be calculated as(1)where is the floor function. For , the formula becomes(2)The number of digits in the number represented in base is given by the Wolfram Language function DigitCount[n, b, d], with DigitCount[n, b] giving a list of the numbers of each digit in . The total number of digits in a number is given by IntegerLength[n, b].The positive integers with distinct base-10 digits are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ... (OEIS A010784). The number of -digit integers is given by(3)(4)(5)(6)where is..

Waring's problem

In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed number of th powers of positive integers, where is any given positive integer and depends only on . Waring originally speculated that , , and . In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacher and Toeplitz 1957, pp. 52-61).In Lagrange's four-square theorem, Lagrange proved that , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy, and Little established , and this was subsequently reduced to by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and..

Reciprocal fibonacci constant

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

Odd number theorem

The sum of the first odd numbers is a square number,A sort of converse also exists, namely the difference of the th and st square numbers is the th odd number, which follows from

Nicomachus's theorem

The th cubic number is a sum of consecutive odd numbers, for example(1)(2)(3)(4)etc. This identity follows from(5)It also follows from this fact that(6)

Pi wordplay

A short mnemonic for remembering the first seven decimal digits of is "How I wish I could calculate pi" (C. Heckman, pers. comm., Feb. 3, 2005). Eight digits are given by "May I have a large container of coffee?" giving 3.1415926 (Gardner 1959; 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). "But I must a while endeavour to reckon right" gives nine correct digits (3.14159265). "May I have a white telephone, or pastel color" (M. Amling, pers. comm., Jul. 31, 2004) also gives nine correct digits.A more substantial mnemonic giving 15 digits (3.14159265358979) is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics," originally due to Sir James Jeans (Gardner 1966, p. 92; Castellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension..

Sierpiński sieve

The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński gasket or Sierpiński triangle. The curve can be written as a Lindenmayer system with initial string "FXF--FF--FF", string rewriting rules "F" -> "FF", "X" -> "--FXF++FXF++FXF--", and angle .The th iteration of the Sierpiński sieve is implemented in the Wolfram Language as SierpinskiMesh[n].Let be the number of black triangles after iteration , the length of a side of a triangle, and the fractional area which is black after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A020857; Wolfram 1984; Borwein and Bailey2003, p. 46).The Sierpiński sieve is produced by the beautiful recurrenceequation(8)where denote bitwise..

Binary carry sequence

The sequence given by the exponents of the highest power of 2 dividing , i.e., the number of trailing 0s in the binary representation of . For , 2, ..., the first few are 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ... (OEIS A007814).Amazingly, this corresponds to one less than the number of disks to be moved at th step in the optimal solution to the tower of Hanoi problem: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, ... (OEIS A001511). The parity of this sequence is given by 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, ... (OEIS A035263) which, amazingly, also corresponds to the accumulation point of cycles through successive bifurcations.

Divisibility tests

In general, an integer is divisible by iff the digit sum is divisible by .Write a positive decimal integer out digit by digit in the form . The following rules then determine if is divisible by another number by examining the congruence properties of its digits. In congruence notation, means that the remainder when is divided by a modulus is . (Note that it is always true that for any base.) 1. All integers are divisible by 1. 2. , so for . Therefore, if the last digit is divisible by 2 (i.e., is even), then so is . 3. , , , ..., (mod 3). Therefore, if the digit sum is divisible by 3, so is (Wells 1986, p. 48). In general, if the sum of any permutation of the digits of in any order is divisible by 3, then so is . 4a. , , ..., (mod 4). So if the last two digits are divisible by 4, then so is . 4b. If is, then so is . 5. , so for . Therefore, if the last digit is divisible by 5 (i.e., is 5 or 0), then so is . 6a. If is divisible by 3 and is even, then is also divisible by 6. 6b. , , ..., (mod 6)...

Carefree couple

Define a carefree couple as a pair of positive integers such that and are relatively prime (i.e., ) and is squarefree. Similarly, define a strongly carefree couple as a pair such that and both and are squarefree, and a weakly carefree couple as a pair such that and at least of one and is squarefree.Let be the number of squarefree pairs, the number of carefree couples, the number of strongly carefree couples, and the number of weakly squarefree couples with , illustrated above.The numbers of squarefree pairs for , 2, ... are 1, 3, 7, 11, 19, 23, 35, 43, 55, ... (OEIS A018805), which has closed forms(1)(2)where is the totient summatory function, is the floor function, and is the Möbius function.The numbers of carefree couples for , 2, ... are 1, 3, 7, 9, 16, 20, 31, 35, 39, ... (OEIS A118258); the numbers of strongly carefree couples are 1, 3, 7, 7, 13, 17, 27, 27, ... (OEIS A118259); and the numbers of weakly carefree couples are 1, 3, 7, 11, 19, 23, 35, 43, 51,..

Joyce sequence

The sequence of numbers giving the number of digits in the three-fold power tower . The values of for , 2, ... are 1, 16, 7625597484987, ... (OEIS A002488; Rossier 1948), so the Joyce sequence is 1, 2, 13, 155, 2185, 36306, ... (OEIS A054382). Laisant (1906) found the term , and Uhler (1947) published the logarithm of this number to 250 decimal places (Wells 1986, p. 208).The sequence is named in honor of the following excerpt from the "Ithaca" chapter of James Joyce's Ulysses: "Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete..

Binet forms

The two recursive sequences(1)(2)with , and , , can be solved for the individual and . They are given by(3)(4)where(5)(6)(7)A useful related identity is(8)Binet's Fibonacci number formula is a special case of the Binet form for corresponding to .

Catalan number

The Catalan numbers on nonnegative integers are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular -gon be divided into triangles if different orientations are counted separately?" (Euler's polygon division problem). The solution is the Catalan number (Pólya 1956; Dörrie 1965; Honsberger 1973; Borwein and Bailey 2003, pp. 21-22), as graphically illustrated above (Dickau).Catalan numbers are commonly denoted (Graham et al. 1994; Stanley 1999b, p. 219; Pemmaraju and Skiena 2003, p. 169; this work) or (Goulden and Jackson 1983, p. 111), and less commonly (van Lint and Wilson 1992, p. 136).Catalan numbers are implemented in the WolframLanguage as CatalanNumber[n].The first few Catalan numbers for , 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (OEIS A000108).Explicit formulas for include(1)(2)(3)(4)(5)(6)(7)where..

Apéry number

Apéry's numbers are defined by(1)(2)(3)where is a binomial coefficient. The first few for , 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).The case of Schmidt's problem expresses these numbers in the form(4)(Strehl 1993, 1994; Koepf 1998, p. 55).They are also given by the recurrence equation(5)with and (Beukers 1987).There is also an associated set of numbers(6)(7)(Beukers 1987), where is a generalized hypergeometric function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828), with no others for (Weisstein, Mar. 8, 2004).The..

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