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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).

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..

A figurate number which is the sum of two consecutivepyramidal numbers,(1)The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (OEIS A005900). The generating function for the octahedral numbers is(2)Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (Dickson 2005, p. 23).A related set of numbers is the number of cubes in the Haűy construction of the octahedron. Each cross section has area(3)where is an odd number, and adding all cross sections gives(4)for an odd number. Re-indexing so that gives(5)the first few values of which are 1, 7, 25, 63, 129, ... (OEIS A001845).These numbers have the generating function(6)

Define as the quantity appearing in Waring's problem, then Euler conjectured thatwhere is the floor function.

Multiply all the digits of a number by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence, and the final 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 an multiplicative persistence of two and a multiplicative digital root of 0. The multiplicative 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, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, ... (OEIS A031346). The smallest numbers having multiplicative persistences of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (OEIS A003001; Wells 1986, p. 78). There is no number with multiplicative persistence (Carmody 2001; updating Wells 1986, p. 78). It is conjectured that the..

A figurate number of the form(1)(2)(3)where is the th triangular number and is a binomial coefficient. These numbers correspond to placing discrete points in the configuration of a tetrahedron (triangular base pyramid). Tetrahedral numbers are pyramidal numbers with , and are the sum of consecutive triangular numbers. The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (OEIS A000292). The generating function for the tetrahedral numbers is(4)Tetrahedral numbers are even, except for every fourthtetrahedral number, which is odd (Conway and Guy 1996).The only numbers which are simultaneously square and tetrahedral are , , and (giving , , and ), as proved by Meyl (1878; cited in Dickson 2005, p. 25).Numbers which are simultaneously triangular and tetrahedral satisfy the binomial coefficient equation(5)the only solutions of which are(6)(7)(8)(9)(10)(OEIS A027568; Avanesov 1966/1967; Mordell1969, p. 258; Guy 1994, p. 147).Beukers..

A solitary number is a number which does not have any friends. Solitary numbers include all primes, prime powers, and numbers for which , where is the greatest common divisor of and and is the divisor function. The first few numbers satisfying are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (OEIS A014567). Numbers such as 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 can also be easily proved to be solitary (Hickerson 2002).Some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large. For example, 24 is friendly because is a friendly pair. However, there exist numbers such as , 45, 48, and 52 which are solitary but for which . It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary,..

The Smarandache function is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that gives the smallest value for a given at which (i.e., divides factorial). For example, the number 8 does not divide , , , but does divide , so .For , 2, ..., is given by 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (OEIS A002034), where it should be noted that Sloane defines , while Ashbacher (1995) and Russo (2000, p. 4) take . The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A046022), which occur at the values where . The incrementally smallest values of relative to are = 1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 3/40, 1/15, 1/16, 1/24, 1/30, ... (OEIS A094404 and A094372), which occur at , 6, 12, 20, 24, 40, 60, 80, 90, 112, 120, 180, ... (OEIS A094371).Formulas exist for immediately computing for special forms of . The simplest cases are(1)(2)(3)(4)(5)where is a prime,..

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 smallest known example is , proved in 1962 by J. Selfridge, but the fate of a number of smaller candidates remains to be determined before this number can be established as the smallest such number. As of 1996, 35 candidates remained (Ribenboim 1996, p. 358), a number which had been reduced to 17 by the beginning of 2002 (Peterson 2003).In March 2002, L. K. Helm and D. A. Norris began a distributed computing effort dubbed "seventeen or bust" to eliminate the remaining candidates. With the aid of collaborators across the globe, this number was reduced to 12 as of December 2003 (Peterson 2003, Helm and Norris). The following table summarizes numbers subsequently found to be prime by "seventeen or bust," leaving only..

Letwhere is the divisor function and is the restricted divisor function, and define the aliquot sequence of byIf the sequence reaches a constant, the constant is known as a perfect number. A number that is not perfect but whose sequence becomes constant is known as an aspiring number. For example, beginning with 25 gives the sequence 25, 6, 6, 6, ..., so 25 is an aspiring number and 6 is a perfect number.The first few aspiring numbers are 25, 95, 119, 143, ... (OEIS A063769). It is not known if 276 is an aspiring number, though it is very unlikely for this to be the case.

An almost perfect number, also known as a least deficient or slightly defective (Singh 1997) number, is a positive integer for which the divisor function satisfies . The only known almost perfect numbers are the powers of 2, namely 1, 2, 4, 8, 16, 32, ... (OEIS A000079).It seems to be an open problem to show that a number is almost perfect only if it is of the form .

A quasiperfect number, called a "slightly excessive number" by Singh (1997), is a "least" abundant number, i.e., one such thatQuasiperfect numbers are therefore the sum of their nontrivial divisors. No quasiperfect numbers are known, although if any exist, they must be greater than and have seven or more distinct prime factors (Hagis and Cohen 1982).

Let be the th Bernoulli number and considerwhere the residues of fractions are taken in the usual way so as to yield integers, for which the minimal residue is taken. Agoh's conjecture states that this quantity is iff is prime. There are no counterexamples less than (S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's conjecture would be a contradiction to Giuga's conjecture, and vice versa.For , 2, ..., the minimal integer residues (mod ) is 0, , , 0, , 0, , 0, , 0, , ... (OEIS A046094).Kellner (2002) provided a short proof of the equivalence of Giuga'sand Agoh's conjectures. The combined conjecture can be described by a sum of fractions.

A nexus number is a figurate number built up of the nexus of cells less than steps away from a given cell. The th -dimensional nexus number is given by(1)(2)where is a binomial coefficient. The symbolic representations and sequences for first few -dimensional nexus numbers are given in the table below.name01unit1odd number2hex number3rhombic dodecahedral number4nexus numberOEIS, , , ...01, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...1A0054081, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...2A0032151, 7, 19, 37, 61, 91, 127, 169, 217, ...3A0059171, 15, 65, 175, 369, 671, 1105, 1695, 2465, ...4A0225211, 31, 211, 781, 2101, 4651, 9031, 15961, ...

The th central fibonomial coefficient is defined as(1)(2)where is a fibonomial coefficient, is a Fibonacci number, is the golden ratio, and is a q-Pochhammer symbol (E. W. Weisstein, Dec. 8, 2009).For , 2, ..., the first few are 1, 6, 60, 1820, 136136, ... (OEIS A003267).

The idempotent numbers are given bywhere is a Bell polynomial and is a binomial coefficient. A table of the first few is given below.A000027A001788A036216A040075A050982A050988A050989112213361442412155809020166240540240301776722835224052542188179213608179207000100856994608612361290247875018144176410101152026244086016078750027216041160

(1)The number triangle illustrated above (OEIS A008949) composed of the partial sums of binomial coefficients,(2)(3)where is a gamma function and is a hypergeometric function.The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Bernoulli triangle.

The fibonomial coefficient (sometimes also called simply the Fibonacci coefficient) is defined by(1)where and is a Fibonacci number. This coefficient satisfies(2)for , where is a Lucas number.The triangle of fibonomial coefficients is given by(3)(OEIS A010048). may be called the central fibonomial coefficient by analogy with the central binomial coefficient.

Pascal's triangle is a number triangle with numbersarranged in staggered rows such that(1)where is a binomial coefficient. The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám. It is therefore known as the Yanghui triangle in China. Starting with , the triangle is(2)(OEIS A007318). Pascal's formula shows that each subsequent row is obtained by adding the two entries diagonally above,(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Pascal's triangle.The first number after the 1 in each row divides all other numbers in that row iff it is a prime.The sums of the number of odd entries in the first rows of Pascal's triangle for , 1, ... are 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, ... (OEIS A006046). It is then..

The Wolstenholme numbers are defined as the numerators of the generalized harmonic number appearing in Wolstenholme's theorem. The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (OEIS A007406).By Wolstenholme's theorem, for prime , where is the th Wolstenholme number. In addition, for prime .The first few prime Wolstenholme numbers are 5, 266681, 40799043101, 86364397717734821, ... (OEIS A123751), which occur at indices , 7, 13, 19, 121, 188, 252, 368, 605, 745, ... (OEIS A111354).

The previous prime function gives the largest prime less than . The function can be given explicitly aswhere is the th prime and is the prime counting function. For , 4, ... the values are 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, ... (OEIS A007917).The previous prime function was implemented in versions of the Wolfram Language prior to 6 as PreviousPrime[n] (after loading the package NumberTheory`NumberTheoryFunctions).Finding the previous prime before gives the largest -digit prime. For , 2, ..., the first few of these are 7, 97, 997, 9973, 99991, 999983, 9999991, 99999989, ... (OEIS A003618). The amounts by which these are less than are 3, 3, 3, 3, 27, 9, 17, 9, 11, 63, ... (OEIS A033873).

Let and be Lucas sequences generated by and , and define(1)Let be an odd composite number with , and with odd and , where is the Legendre symbol. If(2)or(3)for some with , then is called a strong Lucas pseudoprime with parameters .A strong Lucas pseudoprime is a Lucas pseudoprime to the same base. Arnault (1997) showed that any composite number is a strong Lucas pseudoprime for at most 4/15 of possible bases (unless is the product of twin primes having certain properties).

If is prime, then , where is a member of the Perrin sequence 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608). A Perrin pseudoprime is a composite number such that . Several "unrestricted" Perrin pseudoprimes are known, the smallest of which are 271441, 904631, 16532714, 24658561, ... (OEIS A013998).Adams and Shanks (1982) discovered the smallest unrestricted Perrin pseudoprime after unsuccessful searches by Perrin (1899), Malo (1900), Escot (1901), and Jarden (1966). (A 1996 article by Stewart's stating that no Perrin pseudoprimes were then known was incorrect.)Grantham generalized the definition of Perrin pseudoprime with parameters to be an odd composite number for which either 1. and has an S-recurrence relation signature, or 2. and has a Q-recurrence relation signature, where is the Jacobi symbol. All the 55 Perrin pseudoprimes less than have been computed by Kurtz et al. (1986). All have S-recurrence relation signature,..

A pseudoprime which obeys an additional restriction beyond that required for a Frobenius pseudoprime. A number with is a strong Frobenius pseudoprime with respect to iff is a strong pseudoprime with respect to . Every strong Frobenius pseudoprime with respect to is an Euler pseudoprime to the base .Every strong Frobenius pseudoprime with respect to such that is a strong Lucas pseudoprime with parameters . Every strong Frobenius pseudoprime with respect to is an extra strong Lucas pseudoprime to the base .

When and are integers such that , define the Lucas sequence byfor , with and the two roots of . Then define a Lucas pseudoprime as an odd composite number such that , the Jacobi symbol , and .The congruence holds for every prime number , where is a Lucas number. However, some composites also satisfy this congruence. The Lucas pseudoprimes corresponding to the special case of the Lucas numbers are those composite numbers such that . The first few of these are 705, 2465, 2737, 3745, 4181, 5777, 6721, ... (OEIS A005845).The Wolfram Language implements the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test in the function PrimeQ[n].

Let be an elliptic curve defined over the field of rationals having equationwith and integers. Let be a point on with integer coordinates and having infinite order in the additive group of rational points of , and let be a composite natural number such that , where is the Jacobi symbol. Then if is called an elliptic pseudoprime for .

Let be an elliptic pseudoprime associated with , and let with odd and . Then is a strong elliptic pseudoprime when either or for some with .

Let be a monic polynomial of degree with discriminant . Then an odd integer with is called a Frobenius pseudoprime with respect to if it passes a certain algorithm given by Grantham (1996). A Frobenius pseudoprime with respect to a polynomial is then a composite Frobenius probably prime with respect to the polynomial .While 323 is the first Lucas pseudoprime with respect to the Fibonacci polynomial , the first Frobenius pseudoprime is 5777. If , then any Frobenius pseudoprime with respect to is also a Perrin pseudoprime. Grantham (1997) gives a test based on Frobenius pseudoprimes which is passed by composite numbers with probability at most 1/7710.

Consider a Lucas sequence with and . A Fibonacci pseudoprime is a composite number such thatThere exist no even Fibonacci pseudoprimes with parameters and (Di Porto 1993) or (André-Jeannin 1996). André-Jeannin (1996) also proved that if and , then there exists at least one even Fibonacci pseudoprime with parameters and .

A Fermat pseudoprime to a base , written psp(), is a composite number such that , i.e., it satisfies Fermat's little theorem. Sometimes the requirement that must be odd is added (Pomerance et al. 1980) which, for example would exclude 4 from being considered a psp(5).psp(2)s are called Poulet numbers or, less commonly, Sarrus numbers or Fermatians (Shanks 1993). The following table gives the first few Fermat pseudoprimes to some small bases .OEIS-Fermat pseudoprimes2A001567341, 561, 645, 1105, 1387, 1729, 1905, ...3A00593591, 121, 286, 671, 703, 949, 1105, 1541, 1729, ...4A02013615, 85, 91, 341, 435, 451, 561, 645, 703, ...5A0059364, 124, 217, 561, 781, 1541, 1729, 1891, ...If base 3 is used in addition to base 2 to weed out potential composite numbers, only 4709 composite numbers remain . Adding base 5 leaves 2552, and base 7 leaves only 1770 composite numbers.The following table gives the number of Fermat pseudoprimes to various small bases..

A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers. Unfortunately, some authors drop the "composite" requirement, calling any number that passes the specified tests a pseudoprime even if it is prime. Pomerance, Selfridge, and Wagstaff (1980) restrict their use of "pseudoprime" to odd composite numbers."Pseudoprime" used without qualification means Fermat pseudoprime, i.e., a composite number that nonetheless satisfies Fermat's little theorem to some base or set of bases.Carmichael numbers are odd composite numbers that are Fermat pseudoprimes to every base; they are sometimes called absolute pseudoprimes.The following table gives the number of Poulet numbers psp(2), Euler-Jacobi pseudoprimes ejpsp(2), and strong pseudoprimes spsp(2) to the base 2, and Carmichael numbers CN that are smaller than the first few powers of 10 (Guy 1994)...

Given the Lucas sequence and , define . Then an extra strong Lucas pseudoprime to the base is a composite number , where is odd and such that either and , or for some with . An extra strong Lucas pseudoprime is a strong Lucas pseudoprime with parameters . Composite are extra strong pseudoprimes for at most 1/8 of possible bases (Grantham 1997).

The th Ramanujan prime is the smallest number such that for all , where is the prime counting function. In other words, there are at least primes between and whenever . The smallest such number must be prime, since the function can increase only at a prime.Equivalently,Using simple properties of the gamma function, Ramanujan (1919) gave a new proof of Bertrand's postulate. Then he proved the generalization that , 2, 3, 4, 5, ... if , 11, 17, 29, 41, ... (OEIS A104272), respectively. These are the first few Ramanujan primes.The case for all is Bertrand's postulate.

A positive integer is quiteprime iff all primes satisfyAlso define 2 and 3 to be quiteprimes. Then the first few quiteprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 137, ... (OEIS A050260), and the first few primes which are not quiteprimes are 131, 181, 197, 199, 233, 241, 263, 307, 311, 313, 331, 337, 353, 373, 379, ... (OEIS A050261).

A number where the s are distinct primes and such that(1)for , 2, ..., , taken as 1, and with and some fixed integers. For example, is a Zeisel number with since(2)as is since(3)The first few Zeisel numbers are 105, 1419, 1729, 1885, 4505, ... (OEIS A051015), which correspond to constants (1, 2), (4, ), (1, 6), (2, 3), (3, 2), ....

A prime power is a prime or integer power of a prime. A test for a number being a prime power will be implemented in the Wolfram Language as PrimePowerQ[n].The first few prime powers are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, ... (OEIS A000961). The first few prime powers with power are given by 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, ... (OEIS A025475). The number of prime powers (exponents ) up to is given by(Hardy 1999, p. 27).The following table gives prime th powers.OEISprime th powers1A0000402, 3, 5, 7, 11, 13, 17, 19, 23, ...2A0012484, 9, 25, 49, 121, 169, 289, 361, ...3A0300788, 27, 125, 343, 1331, 2197, 4913, ...4A03051416, 81, 625, 2401, 14641, 28561, 83521, ...5A05099732, 243, 3125, 16807, 161051, 371293, ...

A Goldbach number is a positive integer that is the sum of two odd primes (Li 1999). Let (the "exceptional set of Goldbach numbers") denote the number of even numbers not exceeding which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that for every . Li (1999) proved that for sufficiently large ,

A positive integer is a veryprime iff all primes satisfy(1)The weak veryprimes are then 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 103, 107, 137, 157, 173, 227, 347, 487, 773, ... (OEIS A050264), the strong veryprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137, 227, ..., and the very strong veryprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137, ..., with no others in the first primes.

For an integer , let denote the least prime factor of . A pair of integers is called a twin peak if 1. , 2. , 3. For all , implies . A broken-line graph of the least prime factor function resembles a jagged terrain of mountains. In terms of this terrain, a twin peak consists of two mountains of equal height with no mountain of equal or greater height between them. Denote the height of twin peak by . By definition of the least prime factor function, must be prime.Call the distance between two twin peaks (1)Then must be an even multiple of ; that is, where is even. A twin peak with is called a -twin peak. Thus we can speak of -twin peaks, -twin peaks, etc. A -twin peak is fully specified by , , and , from which we can easily compute .The set of -twin peaks is periodic with period , where is the primorial of . That is, if is a -twin peak, then so is . A fundamental -twin peak is a twin peak having in the fundamental period . The set of fundamental -twin peaks is symmetric with respect to..

A zerofree number is called right truncatable if and all numbers obtained by successively removing the rightmost digits are prime. There are exactly 83 right truncatable primes in base 10. The first few are 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, ... (OEIS A024770), the largest being the 8-digit number (Angell and Godwin 1977). The numbers of -digit right prime strings for , 2, ..., 8 are 4, 9, 14, 16, 15, 12, 8, and 5 (OEIS A050986; Rivera puzzle 70).Similarly, call a number left truncatable if and all numbers obtained by successively removing the leftmost digit are prime. There are exactly 4260 left truncatable primes in base 10 when the digit zero is not allowed. The first few are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, ... (OEIS A024785), with the largest being the 24-digit number (Angell and Godwin 1977). The numbers of -digit left truncatable primes for , 2, ... 24 are 4,..

A Størmer number is a positive integer for which the greatest prime factor of is at least . Every Gregory number can be expressed uniquely as a sum of s where the s are Størmer numbers. The first few Størmer numbers are given by Conway and Guy (1996) and Todd (1949) and are given by , 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, ... (OEIS A005528), corresponding to greatest prime factors 2, 5, 17, 13, 37, 41, 101, 61, 29, ... (OEIS A005529).

The cuban primes, named after differences between successive cubic numbers, have the form . The first few are 7, 19, 37, 61, 127, 271, ... (OEIS A002407), which are also the prime hex numbers. They correspond to indices , 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, ... (OEIS A002504; Cunningham 1912).The numbers of cuban primes less than 1, 10, , ... are 0, 1, 4, 11, 28, 64, 173, 438, 1200, ... (OEIS A113478), which is well-approximated byCuban primes are cyclotomic in nature, being the evaluation of the third homogeneous cyclotomic polynomial, , at values and . The form therefore can only have primitive factors of the form . Also, by construction, 2 and 3 are excluded as non-primitive factors. Therefore, this form has a slightly higher density than would arbitrary numbers of the same size (P. Carmody, pers. comm., Jan. 8, 2006). ..

A Chen prime is a prime number for which is either a prime or semiprime. Chen primes are named after Jing Run Chen who proved in 1966 that there are infinitely many such primes (Chen's theorem).The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A109611). The first primes that are not Chen primes are 43, 61, 73, 79, 97, 103, 151, ... (OEIS A102540).The lesser of any twin prime is always a Chen prime. Apart from twin prime records, the largest known Chen prime known as of Oct. 2005 was(https://primes.utm.edu/primes/page.php?id=75857),which has 70301 digits.There are infinitely many cases of 3 Chen primes in arithmetic progression (Green and Tao 2005). The following 3074-digit case produces Chen primes for , 1, 2, where denotes the primorial:

There exist infinitely many odd integers such that is composite for every . Numbers with this property are called Riesel numbers, while analogous numbers with the minus sign replaced by a plus are called Sierpiński numbers of the second kind.The smallest known Riesel number is , but as of Jan. 2014, there remain 52 smaller candidates which generate only composite numbers for all which have been checked (Ribenboim 1996, p. 358; Ballinger and Keller; Keller): 2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 273809, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 402539, 409753, 444637, 470173, 474491, 477583, 485557, 494743, and 502573.The problem of proving or disproving that is the smallest..

For every , let be the set of composite numbers such that if , (where GCD is the greatest common divisor), then .Special cases include , which is the set of Carmichael numbers, and , which gives the D-numbers.Makowski (1962/1963) proved that there are infinitely many members of for . The following table summarized Knödel numbers for small .OEIS1A002997561, 1105, 1729, 2465, 2821, 6601, 8911, ...2A0509904, 6, 8, 10, 12, 14, 22, 24, 26, 30, ...3A0335539, 15, 21, 33, 39, 51, 57, 63, 69, 87, ...4A0509926, 8, 12, 16, 20, 24, 28, 40, 44, 48, ...5A05099325, 65, 85, 145, 165, 185, 205, ...

A Brier number is a number that is both a Riesel number and a Sierpiński number of the second kind, i.e., a number such that for all , the numbers and are composite. The first few are 878503122374924101526292469, 3872639446526560168555701047, 623506356601958507977841221247, ... (OEIS A076335).

An integer is called a jumping champion if is the most frequently occurring difference between consecutive primes (Odlyzko et al. 1999). This term was coined by J. H. Conway in 1993. There are occasionally several jumping champions in a range. The scatter plots above show the jumping champions for small , and the ranges of number having given jumping champion sets are summarized in the following table.131, 2527-100, 103-106, 109-112, ...2, 4101-102, 107-108, 113-130, ...4131-138, ...2, 4, 6179-180, 467-490, ...2, 6379-388, 421-432, ...6389-420, ...Odlyzko et al. (1999) give a table of jumping champions for , consisting mainly of 2, 4, and 6. 6 is the jumping champion up to about , at which point 30 dominates. At , 210 becomes champion, and subsequent primorials are conjectured to take over at larger and larger . Erdős and Straus (1980) proved that the jumping champions tend to infinity under the assumption of a quantitative..

A palindromic number is a number (in some base ) that is the same when written forwards or backwards, i.e., of the form . The first few palindromic numbers are therefore are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ... (OEIS A002113). The number of palindromic numbers less than a given number are illustrated in the plot above.A number can be tested to see if it is palindromic in the Wolfram Language using PalindromeQ[n].The numbers of palindromic numbers less than 10, , , ... are 9, 18, 108, 198, 1098, 1998, 10998, ... (OEIS A050250). This sequence is given by the closed-form formula(1)Banks et al. (2004) proved that almost all palindromes (in any base) are composite,with the precise statement being(2)where is the number of palindromic primes and is the number of palindromic numbers .The sum of the reciprocals of the palindromic numbers converges to a constant (OEIS A118031; Rivera), where the value has been computed..

Define an emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, because 15 reversed is 51 and both 15 and 51 are semiprimes (i.e., and ). A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (OEIS A097393). A binary plot of the semirpimes is illustrated above.The numbers of semirpimes less than for , 2, ... are 0, 10, 90, 898, 7200, 60732, ... (OEIS A097394).

The initially palindromic numbers 1, 121, 12321, 1234321, 123454321, ... (OEIS A002477). For the first through ninth terms, the sequence is given by the generating function(1)(Plouffe 1992, Sloane and Plouffe 1995).The definition of this sequence is slightly ambiguous from the tenth term on, but the most common convention follows from the following observation. The sequences of consecutive and reverse digits and , respectively, are given by(2)(3)for , so the first few Demlo numbers are given by(4)(5)But, amazingly, this is just the square of the th repunit , i.e.,(6)for , and the squares of the first few repunits are precisely the Demlo numbers: , , , ... (OEIS A002275 and A002477). It is therefore natural to use (6) as the definition for Demlo numbers with , giving 1, 121, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ....The equality for also follows immediately from schoolbook multiplication, as illustrated..

The Wythoff array is an interspersion array that can be constructed by beginning with the Fibonacci numbers in the first row and then building up subsequent rows by iteratively adding , where or 1 is the smallest offset producing an initial term that has not occurred in a previous row. This process gives the array(1)Read by skew diagonals from lower left to upper right, this gives the sequence 1; 4, 2; 6, 7, 3; ... (OEIS A083412), while read by skew diagonals from upper right to lower left, this gives 1; 2, 4; 3, 7, 6; ... (OEIS A035513).The first column is given by 1, 4, 6, 9, 12, 14, 17, ... (OEIS A003622), with the initial term of the th row given by(2)(3)with is the golden ratio. Rows numbered , i.e., 2, 5, 7, 10, 13, ... (OEIS A001950) have offset , while rows numbers , i.e., 1, 3, 4, 6, 8, ... (OEIS A000201) have .The element can be given explicitly by(4)..

The Leibniz harmonic triangle is the number trianglegiven by(1)(OEIS A003506), where each fraction is the sum of numbers below it and the initial and final entries in the th row are given by .The terms are given by the recurrences(2)(3)and explicitly by(4)(5)where is a binomial coefficient.The denominators in the second diagonals are the pronic numbers 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, ... (OEIS A002378). A sorted list of all possible denominators in the triangle is given by 6, 12, 20, 30, 42, 56, 60, 72, 90, 105, 110, ... (OEIS A007622).The row sums are given by 1, 1, 5/6, 2/3, 8/15, 13/30, 151/420, ... (OEIS A046878 and A046879). The sums of the denominators in the th row are given by , giving the first few as 1, 4, 12, 32, 80, 192, 448, ... (OEIS A001787).

The triangle of numbers given by(1)and the recurrence relation(2)for , where are shifted Eulerian numbers, i.e.,(3)(4)(OEIS A008292). Note that the rows sum to the successive factorials , , , , ....The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Euler's number triangle.Amazingly, the Z-transform of is the generator for the first rows of Euler's number triangle, when the th term of the transform is first cleared of its denominator by multiplying through by . For example,(5)

Clark's triangle is a number triangle created by setting the vertex equal to 0, filling one diagonal with 1s, the other diagonal with multiples of an integer , and filling in the remaining entries by summing the elements on either side from one row above. The illustration above shows Clark's triangle for (OEIS A090850).Call the first column and the last column so that(1)(2)then use the recurrence relation(3)to compute the rest of the entries. The result is given analytically by(4)where is a binomial coefficient (M. Alekseyev, pers. comm., Aug. 10, 2005).The interesting part is that if is chosen as the integer, then and simplify to(5)(6)which are consecutive cubes and nonconsecutive squares .The sum of the th row for is given by(7)(M. Alekseyev, pers. comm., Aug. 10, 2005).The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Clark's triangle with..

A monotone triangle (also called a strict Gelfand pattern or a gog triangle) of order is a number triangle with numbers along each side and the base containing entries between 1 and such that there is strict increase across rows and weak increase diagonally up or down to the right. There is a bijection between monotone triangles of order and alternating sign matrices of order obtained by letting the th row of the triangle equal the positions of 1s in the sum of the first rows of an alternating sign matrix, as illustrated below.(1)(2)(3)(4)(5)(6)

A triangle with rows containing the numbers that begins with 1, ends with , and such that the sum of each two consecutive entries being a prime. Rows 2 to 6 are unique,(OEIS A051237) but there are multiple possibilities starting with row 7. For example, the two possibilities for row 7 are and . The number of possible rows ending with , 2, ..., are 0, 1, 1, 1, 1, 1, 2, 4, 7, 24, 80, ... (OEIS A036440).

A number triangle of order with entries 1 to such that entries are nondecreasing across rows and down columns and all entries in column are less than or equal to . An example isMagog triangles are in 1-to-1 correspondence with totally symmetric self-complementaryplane partitions.

The Bell triangle, also called Aitken's array or the Peirce triangle (Knuth 2005, p. 28), is the number triangle obtained by beginning the first row with the number one, and beginning subsequent rows with last number of the previous row. Rows are filled out by adding the number in the preceding column to the number above it (OEIS A011971). The Bell numbers 1, 1, 2, 5, 15, ... (OEIS A000110) are then given as the values in the first column.The name "Bell triangle" was suggested to Gardner by J. Shallit. A reflected version is sometimes also considered (Knuth 2005, p. 28).The sums of the numbers in rows arewhere is a Stirling number of the second kind, giving the first few for , 2, ... as 1, 3, 10, 37, 151, ... (OEIS A005493).

(1)Losanitsch's triangle (OEIS A034851) is a number triangle for which each term is the sum of the two numbers immediately above it, except that, numbering the rows by , 1, 2, ... and the entries in each row by , 1, 2, ..., , are given by the recurrence equations(2)where is a binomial coefficient. can be written in closed form as(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Losanitsch's triangle.The row sums of Losanitsch's triangle are(4)the first few terms of which are 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).

A Euclidean number is a number which can be obtained by repeatedly solving the quadratic equation. Euclidean numbers, together with the rational numbers, can be constructed using classical geometric constructions. However, the cases for which the values of the trigonometric functions sine, cosine, tangent, etc., can be written in closed form involving square roots of real numbers are much more restricted.

A number having generating function(1)(2)For , 2, ..., the denominators are 48, 5760, 362880, 19353600, ... (OEIS A057868).It has closed form(3)(4)and , where is a Bernoulli number and a gamma function.

A Meeussen sequence is an increasing sequence of positive integers (, , ...) such that , every nonnegative integer is the sum of a subset of the , and each integer is the sum of a unique such subset. Cook and Kleber (2000) show that Meeussen sequences are isomorphic to tournament sequences.

A tournament sequence is an increasing sequence of positive integers (, , ...) such that and . Cook and Kleber (2000) show that Meeussen sequences are isomorphic to tournament sequences.

A Cullen number is a number of the formThe first few are 3, 9, 25, 65, 161, 385, ... (OEIS A002064).Cullen numbers are divisible by if is a prime of the form .The first few prime Cullen numbers are numbers 3, 393050634124102232869567034555427371542904833, ... (OEIS A050920), corresponding to , 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... (OEIS A005849; Caldwell). As of Nov. 2015, it is known that there are no other Cullen primes for (PrimeGrid).

The numbers defined by the recurrence relationwith . The first few values for , 1, 2, ... are 1, 3, 3, 4, 7, 7, 7, 9, 9, 10, 13, ... (OEIS A007448).Conway and Guy (1996, p. 15) note that is the alphabetically first prime number in the American system of large number terminology, and term this "Knuth's number."

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.All rational numbers are constructible, and all constructible numbers are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins 1996, p. 136).In particular, let be the field of rationals. Now construct an extension field of constructible numbers by the adjunction of , where is in , but is not, consisting of all numbers of the form , where . Next, construct an extension field of by the adjunction of , defined as the numbers , where , and is a number in for which does not lie in . Continue the process times. Then constructible numbers are precisely those..

The pseudosmarandache function is the smallest integer such thatis divisible by . The values for , 2, ... are 1, 3, 2, 7, 4, 3, 6, 15, 8, 4, ... (OEIS A011772; Kashihara 1996; Russo 2000, p. 4).

A Proth number is a number of the form for odd , a positive integer, and . The condition is needed since otherwise, every odd number would be a Proth number. The first few Proth numbers are 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, ... (OEIS A080075).The Cullen numbers are a special case of the Proth numbers with (and the inequality restriction dropped). The Fermat numbers are a special case of the Proth numbers with .

A binomial number is a number of the form , where , and are integers. Binomial numbers can be factored algebraically as(1)for all ,(2)for odd, and(3)for all positive integers . For example,(4)(5)(6)(7)(8)(9)(10)(11)(12)and(13)(14)(15)(16)(17)(18)(19)(20)(21)Rather surprisingly, the number of factors of with and symbolic and a positive integer is given by , where is the number of divisors of and is the divisor function. The first few terms are therefore 1, 2, 2, 3, 2, 4, 2, ... (OEIS A000005).Similarly, the number of factors of is given by , where is the number of odd divisors of and is the odd divisor function. The first few terms are therefore 1, 1, 2, 1, 2, 2, 2, 1,... (OEIS A001227).In 1770, Euler proved that if , then every odd factor of(22)is of the form . (A number of the form is called a Fermat number.)If and are primes, then(23)is divisible by every prime factor of not dividing ...

The numbers , where is a Hermite polynomial, may be called Hermite numbers. For , 1, ..., the first few are 1, 0, , 0, 12, 0, , 0, 1680, 0, ... (OEIS A067994). They are given explicitly by(1)(2)As a result of the ratio always being divisible by for , the only prime Hermite number is .The Hermite numbers are related to the Hermite polynomials by(3)where , and(4)where .

An NSW number (named after Newman, Shanks, and Williams) is an integer that solves the Diophantine equation(1)In other words, the NSW numbers index the diagonals of squares of side length having the property that the squares of the diagonal equals one plus a square number . Such numbers were called "rational diagonals" by the Greeks (Wells 1986, p. 70). The name "NSW number" derives from the names of the authors of the paper on the subject written by Newman et al. (1980/81).The first few NSW numbers are therefore , 7, 41, 239, 1393, ... (OEIS A002315), which correspond to square side lengths , 5, 29, 169, 985, 5741, 33461, 195025, ... (OEIS A001653). The values indexed by and therefore give 2, 50, 1682, 57122, ... (OEIS A088920).Taking twice the NSW numbers gives the sequence 2, 14, 82, 478, 2786, 16238, ... (OEIS A077444), which is exactly every other Pell-Lucas number.The first few prime NSW numbers are , 41, 239, 9369319,..

A number given by the generating function(1)It satisfies , , and even coefficients are given by(2)(3)where is a Bernoulli number and is an Euler polynomial.The first few Genocchi numbers for , 4, ... are , 1, , 17, , 2073, ... (OEIS A001469).The first few prime Genocchi numbers are and 17, which occur for and 8. There are no others with (Weisstein, Mar. 6, 2004). D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.

The integer sequence 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, ... (OEIS A005044) given by the coefficients of the Maclaurin series for(1)A binary plot of the first few terms in the sequence isillustrated above.Closed forms include(2)(3)(4)where is the floor function.The number of different triangles which have integral sides and perimeter is given by(5)(6)(7)where and are partition functions, with giving the number of ways of writing as a sum of terms, is the nearest integer function, and is the floor function (Jordan et al. 1979, Andrews 1979, Honsberger 1985). Strangely enough, for , 4, ... is precisely Alcuin's sequence.

The exponential factorial is defined by the recurrencerelation(1)where . The first few terms are therefore(2)(3)(4)(5)... (OEIS A049384). The term has digits.The exponential factorial is therefore a kind of "factorial powertower."The sum of the reciprocals of the exponential factorials is given by(6)(7)(OEIS A080219). This sum is a Liouvillenumber and is therefore transcendental.

Two fractions are said to be adjacent if their difference has a unit numerator. For example, 1/3 and 1/4 are adjacent since , but and are not since . Adjacent fractions can be adjacent in a Farey sequence.

There are several results known as the Morgado identity. The first is(1)where is a Fibonacci number and is a Lucas number (Morgado 1987, Dujella 1995).A second Morgado identity is satisfied by generalized Fibonacci numbers ,(2)where(3)(4)(Morgado 1987, Dujella 1996).

A number written as a sum of nonconsecutive Fibonaccinumbers,where are 0 or 1 andEvery positive integer can be written uniquelyin such a form.

The tribonacci numbers are a generalization of the Fibonacci numbers defined by , , , and the recurrence equation(1)for (e.g., Develin 2000). They represent the case of the Fibonacci n-step numbers.The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does..

The Jacobsthal numbers are the numbers obtained by the s in the Lucas sequence with and , corresponding to and . They and the Jacobsthal-Lucas numbers (the s) satisfy the recurrence relation(1)The Jacobsthal numbers satisfy and and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (OEIS A001045). The Jacobsthal-Lucas numbers satisfy and and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, ... (OEIS A014551). The properties of these numbers are summarized in Horadam (1996).Microcontrollers (and other computers) use conditional instructions to change the flow of execution of a program. In addition to branch instructions, some microcontrollers use skip instructions which conditionally bypass the next instruction. This winds up being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 on 5 bits, 21 on 6 bits, 43 on 7 bits, 85 on 8 bits, ..., which are exactly the Jacobsthal numbers (Hirst 2006).The Jacobsthal..

The tetranacci numbers are a generalization of the Fibonacci numbers defined by , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers. The first few terms for , 1, ... are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (OEIS A000078).The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... (OEIS A104534), corresponding to 2, 29, 401, 773, 5350220959, ... (OEIS A104535), with no others for (E. W. Weisstein, Mar. 21, 2009).An exact expression for the th tetranacci number for can be given explicitly by(2)where the three additional terms are obtained by cyclically permuting , which are the four roots of the polynomial(3)Alternately,(4)This can be written in slightly more concise form as(5)where is the th root of the polynomial(6)and and are in the ordering of the Wolfram Language's Root object.The tetranacci numbers..

The hexanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ... (OEIS A001592).An exact formula for the th hexanacci number can be given explicitly in terms of the six roots of(2)as(3)The ratio of adjacent terms tends to the positive root of , namely 1.98358284342... (OEIS A118427), sometimes called the hexanacci constant.

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given byso the next term is 1345.Applied to 1, the sequence gives 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).Conway conjectured that an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (Guy 2004, p. 404).The lengths of the cycles obtained by starting with , 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611), where a 0 indicates that the sequence diverges.The following table summarizes the first few values of leading to a period of length . There are no..

An integer such that if , then , is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, ... (OEIS A001694). Powerful numbers are always of the form for .The numbers of powerful numbers , , , ... are given by 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, ... (OEIS A118896).Golomb (1970) showed that the sum over the reciprocals of the powerful numbers is given by(1)(2)(OEIS A082695), where is the Riemann zeta function.Not every natural number is the sum of two powerful numbers, but Heath-Brown (1988) has shown that every sufficiently large natural number is the sum of at most three powerful numbers. There are infinitely many pairs of consecutive powerful numbers, the first few being (8, 9), (288, 289), (675, 676), (9800, 9801), ... (OEIS A060355 and A118893).Erdős (1975/1965) conjectured that there do not exist three consecutive powerful numbers...

The conjecture due to Pollock (1850) that every number is the sum of at most five tetrahedral numbers (Dickson 2005, p. 23; incorrectly described as "pyramidal numbers" and incorrectly dated to 1928 in Skiena 1997, p. 43). The conjecture is almost certainly true, but has not yet been proven.The numbers that are not the sum of tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (OEIS A000797) of 241 terms, with being almost certainly the last such number.

A pair of primes that sum to an even integer are known as a Goldbach partition (Oliveira e Silva). Letting denote the number of Goldbach partitions of without regard to order, then the number of ways of writing as a sum of two prime numbers taking the order of the two primes into account is(1)The Goldbach conjecture is then equivalent to the statement that or, equivalently, that , for every even integer .A plot of , sometimes known as Goldbach's comet, for up to 2000 is illustrated above.The following table summarizes the values of several variants of for , 4, ....partition typeOEISvalues 1 or primeA0010311, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, ... primeA0459170, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ... odd primeA0023750, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ...Various fractal properties have been observed in Goldbach'spartition (Liang et al. 2006)...

Sloane's1A000027234567892A002993491234683A002994826123574A097408182612465A097409321371356A097410674141257A097411121728248A097412266315149A0974135121141310A09741415196213Consider the leftmost (i.e., most significant) decimal digit of the numbers , , ..., . Then what are the patterns of digits occurring in the table for , 2, ... (King 1994)? For example,1. Will the digit 9 ever occur in the column? The answer is "yes," in particular at values , 63, 73, 83, 93, 156, 166, 176, ... (OEIS A097415. This problem appears in Avez (1966, p. 37), where it is attributed to Gelfand. 2. Will the row "23456789" ever appear for ? None does for . If so, will it have a frequency? If so, will the frequency be rational or irrational? 3. Will a row of all the same digit occur? No such example occurs for . 4. Will the decimal expansion of an 8-digit prime ever occur? (The answer is "yes," in particular at values , 11,..

Apply the 196-algorithm, which consists of taking any positive integer of two digits or more, reversing the digits, and adding to the original number. Now sum the two and repeat the procedure with the sum. Of the first numbers, only 251 do not produce a palindromic number in steps (Gardner 1979).It was therefore conjectured that all numbers will eventually yield a palindromic number. However, the conjecture has been proven false for bases which are a power of 2, and seems to be false for base 10 as well. Among the first numbers, numbers apparently never generate a palindromic number (Gruenberger 1984). The first few are 196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, ... (OEIS A006960).It is conjectured, but not proven, that there are an infinite number of palindromic primes. With the exception of 11, palindromic primes must have an odd number of digits...

Define the abundancy of a positive integer as(1)where is the divisor function. Then a pair of distinct numbers is a friendly pair (and is said to be a friend of ) if their abundancies are equal:(2)For example, (4320, 4680) is a friendly pair since , , and(3)(4)Another example is , which has index 5/2. The first few friendly pairs, ordered by smallest maximum element are (6, 28), (30, 140), (80, 200), (40, 224), (12, 234), (84, 270), (66, 308), ... (OEIS A050972 and A050973).Friendly triples and higher-order tuples are also possible. Friendly triples include (2160, 5400, 13104), (9360, 21600, 23400), and (4320, 4680, 26208), friendly quadruples include (6, 28, 496, 8128), (3612, 11610, 63984, 70434), (3948, 12690, 69936, 76986), and friendly quintuples include (84, 270, 1488, 1638, 24384), (30, 140, 2480, 6200, 40640), (420, 7440, 8190, 18600, 121920).Numbers that have friends are called friendly numbers, and numbers that do not have friends..

An untouchable number is a positive integer that is not the sum of the proper divisors of any number. The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114). Erdős has proven that there are infinitely many.It is thought that 5 is the only odd untouchable number. This would follow from a very slightly stronger version of the Goldbach conjecture, namely the conjecture that every even integer is the sum of two distinct primes. Suppose is an odd number greater than 7. Then by the conjecture, and so the proper divisors of , which are 1, , and , sum to , and so is not untouchable. 1, 3 and 7 are not untouchable, being the sum of the proper divisors of 2, 4, and 8, respectively. That leaves 5 as the only odd untouchable number (F. Adams-Watters, pers. comm., Aug. 4, 2006)...

Define the harmonic mean of the divisors of where is the divisor function (the number of divisors of ).For , 2, ..., the values of are then 1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, 27/13, 20/9, ... (OEIS A099377 and A099378).If is a perfect number, is an integer.Ore conjectured that if is odd, then is not an integer. This implies that no odd perfect numbers exist.

A friendly number is a number that is a member of a friendly pair or a higher-order friendly -tuple. Numbers that are not friendly are said to be solitary. There are some numbers that can easily be proved to be solitary, but the status of numbers 10, 14, 15, 20, and many others remains unknown (Hickerson 2002). The numbers known to be friendly are given by 6, 12, 24, 28, 30, 40, 42, 56, 60, ... (OEIS A074902).Friendly numbers have a positive density.

A polygonal number is a type of figurate number that is a generalization of triangular, square, etc., to an -gon for an arbitrary positive integer. The above diagrams graphically illustrate the process by which the polygonal numbers are built up. Starting with the th triangular number , then(1)Now note that(2)gives the th square number,(3)gives the th pentagonal number, and so on. The general polygonal number can be written in the form(4)(5)where is the th -gonal number (Savin 2000). For example, taking in (5) gives a triangular number, gives a square number, etc.Polygonal numbers are implemented in the WolframLanguage as PolygonalNumber.Call a number -highly polygonal if it is -polygonal in or more ways out of , 4, ... up to some limit. Then the first few 2-highly polygonal numbers up to are 1, 6, 9, 10, 12, 15, 16, 21, 28, (OEIS A090428). Similarly, the first few 3-highly polygonal numbers up to are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (OEIS..

The heptanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ... (OEIS A066178).An exact formula for the th heptanacci number can be given explicitly in terms of the seven roots of(2)as(3)The ratio of adjacent terms tends to the real root of , namely 1.99196419660... (OEIS A118428), sometimes called the heptanacci constant.

Consider the Fibonacci-like recurrence(1)where , , and each sign is chosen independently and at random with probability 1/2. Surprisingly, Viswanath (2000) showed that(2)(OEIS A078416) with probability one. This constantis sometimes known as Viswanath's constant.Considering the more general recurrence(3)the limit(4)exists for almost all values of . The critical value such that is given by(5)(OEIS A118288) and is sometimes known as theEmbree-Trefethen constant.Since Fibonacci numbers can be computed as products of Fibonacci Q-matrices, this same constant arises in the iterated multiplication of certain pairs of random matrices (Bougerol and Lacrois 1985, pp. 11 and 157).

A generalization of the Fibonacci numbers defined by and the recurrence relation(1)These are the sums of elements on successive diagonals of a left-justified Pascal's triangle beginning in the leftmost column and moving in steps of up and 1 right. The case equals the usual Fibonacci number. These numbers satisfy the identities(2)(3)(4)(5)(Bicknell-Johnson and Spears 1996). For the special case ,(6)Bicknell-Johnson and Spears (1996) give many further identities.Horadam (1965) defined the generalized Fibonacci numbers as , where , , , and are integers, , , and for . They satisfy the identities(7)(8)(9)(10)where(11)(12)(Dujella 1996). The final above result is due to Morgado (1987) and is called themorgado identity.Another generalization of the Fibonacci numbers is denoted . Given and , define the generalized Fibonacci number by for ,(13)(14)(15)where the plus and minus signs alternate...

The fibonorial , also called the Fibonacci factorial, is defined aswhere is a Fibonacci number. For , 2, ..., the first few fibonorials are 1, 1, 2, 6, 30, 240, 3120, 65520, ... (OEIS A003266).The fibonorials are asymptotic towhere is the Fibonacci factorial constant and is the golden ratio.The first few values of such that is prime are given by 4, 5, 6, 7, 8, 14, 15, ... (OEIS A059709), with no others less than 500.The first few values of such that is prime are given by 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (OEIS A053408), with no others less than 500.

There are two identities known as Catalan's identity.The first iswhere is a Fibonacci number. Letting gives Cassini's Identity.The second is the trivariate identity on partition of cubes into a sum of three squares given by

For the th Fibonacci number,This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Wells 1986, p. 62). It is a special case of Catalan's identity with .

The pentanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... (OEIS A001591).The ratio of adjacent terms tends to the real root of , namely 1.965948236645485... (OEIS A103814), sometimes called the pentanacci constant.An exact formula for the th pentanacci number can be given explicitly in terms of the five roots of(2)as(3)The pentanacci numbers have generating function(4)

Binet's formula is a special case of the Binet form with , corresponding to the th Fibonacci number,(1)(2)where is the golden ratio. It was derived by Binet in 1843, although the result was known to Euler, Daniel Bernoulli, and de Moivre more than a century earlier.

A number for which the harmonic mean of the divisors of , i.e., , is an integer, where is the number of positive integer divisors of and is the divisor function. For example, the divisors of are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, giving(1)(2)(3)so 140 is a harmonic divisor number. Harmonic divisor numbers are also called Ore numbers. Garcia (1954) gives the 45 harmonic divisor numbers less than . The first few are 1, 6, 28, 140, 270, 496, ... (OEIS A001599).For distinct primes and , harmonic divisor numbers are equivalent to even perfect numbers for numbers of the form . Mills (1972) proved that if there exists an odd positive harmonic divisor number , then has a prime-power factor greater than .Another type of number called "harmonic" is the harmonicnumber.

Let be the divisor function of . Then two numbers and are a quasiamicable pair ifThe first few are (48, 75), (140, 195), (1050, 1925), (1575, 1648), ... (OEIS A005276). Quasiamicable numbers are sometimes called betrothed numbers or reduced amicable pairs.

Any composite number with for all prime divisors of . is a Giuga number iff(1)where is the totient function and iff(2) is a Giuga number iff(3)where is a Bernoulli number and is the totient function. Every counterexample to Giuga's conjecture is a contradiction to Agoh's conjecture and vice versa. The smallest known Giuga numbers are 30 (3 factors), 858, 1722 (4 factors), 66198 (5 factors), 2214408306, 24423128562 (6 factors), 432749205173838, 14737133470010574, 550843391309130318 (7 factors),244197000982499715087866346, 554079914617070801288578559178(8 factors), ... (OEIS A007850).It is not known if there are an infinite number of Giuga numbers. All the above numbers have sum minus product equal to 1, and any Giuga number of higher order must have at least 59 factors. The smallest odd Giuga number must have at least nine prime factors...

A pseudoperfect number, sometimes also called a semiperfect number (Benkoski 1972, Butske et al. 1999), is a positive integer such as which is the sum of some (or all) of its proper divisors. Identifying pseudoperfect numbers is therefore equivalent to solving the subset sum problem.A pseudoperfect number which is the sum of all its proper divisors is called a perfect number.The first few pseudoperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (OEISA005835).Every positive integer is pseudoperfect sinceand , , and are all proper divisors of . Every multiple of a pseudoperfect number is pseudoperfect, as are all numbers for and a prime between and (Guy 1994, p. 47).A pseudoperfect number cannot be deficient (or therefore prime). Rare abundant numbers which are not pseudoperfect are called weird numbers...

A proper factor of a positive integer is a factor of other than 1 or (Derbyshire 2004, p. 32). For example, 2 and 3 are positive proper factors of 6, but 1 and 6 are not.Compare with the term proper divisor, which means a factor of other than (but including 1).

A positive proper divisor is a positive divisor of a number , excluding itself. For example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself is not. The number of proper divisors of is therefore given bywhere is the divisor function. For , 2, ..., is therefore given by 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, ... (OEIS A032741). The largest proper divisors of , 3, ... are 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, ... (OEIS A032742).The term "proper divisor" is sometimes used to include negative integer divisors of a number (excluding ). Using this definition, , , , 1, 2, and 3 are the proper divisors of 6, while and 6 are the improper divisors.To make matters even more confusing, the proper divisor is often defined so that and 1 are also excluded. Using this alternative definition, the proper divisors of 6 would then be , , 2, and 3, and the improper divisors would be , , 1, and 6...

In 1657, Fermat posed the problem of finding solutions toand solutions towhere is the divisor function (Dickson 2005).The first few solutions to are , (7, 20), (751530, 1292054400) (OEIS A008849 and A048948) .... Lucas stated that there are an infinite number of solutions (Dickson 2005, p. 56), but only solutions up to the fourth are known to be complete.The first few solutions to are , (43098, 1729), ... (OEIS A008850 and A048949), with only solutions up to the second known to be complete.

A unitary perfect number is a number which is the sum of its unitary divisors with the exception of itself. There are no odd unitary perfect numbers, and it has been conjectured that there are only a finite number of even ones. The first few are 6, 60, 90, 87360, 146361946186458562560000, ... (OEIS A002827).

An even perfect number is perfect number that is even, i.e., an even number whose sum of divisors (including itself) equals .All known perfect numbers are even, and Ochem and Rao (2012) have shown that any odd perfect number must be larger than .

A number which is an integer multiple of the sum of its unitary divisors is called a unitary -multiperfect number. There are no odd unitary multiperfect numbers.

The numbers and are an amicable pair if the three integers(1)(2)(3)are all prime numbers for some positive integer satisfying (Dickson 2005, p. 42). However, there are many amicable pairs which do not satisfy Euler's rule, so it is a sufficient but not necessary condition for amicability. Euler's rule is a generalization of Thâbit ibn Kurrah rule.The first few for which Euler's rule is satisfied are , , , , , ... (OEIS A094445 and A094446), with no others for , corresponding to the triples , (23, 47, 1151), (191, 383, 73727), ..., giving the amicable pairs (220, 284), (17296, 18416), (9363584, 9437056), ....

An integer which is a product of distinct primes and which satisfies(Butske et al. 1999). The first few are 2, 6, 42, 1806, 47058, ... (OEIS A054377).The similar equationarises in the definition of Giuga numbers.

A number is practical if for all , is the sum of distinct proper divisors of . Defined in 1948 by A. K. Srinivasen. All even perfect numbers are practical. The numberis practical for all , 3, .... The first few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, ... (OEIS A005153). G. Melfi has computed twins, triplets, and 5-tuples of practical numbers. The first few 5-tuples are 12, 18, 30, 198, 306, 462, 1482, 2550, 4422, ....

Thâbit ibn Kurrah's rules is a beautiful result of Thâbit ibn Kurrah dating back to the tenth century (Woepcke 1852; Escott 1946; Dickson 2005, pp. 5 and 39; Borho 1972). Take and suppose that(1)(2)(3)are all prime. Then are an amicable pair, where is sometimes called a Thâbit ibn Kurrah number. This form was rediscovered by Fermat in 1636 and Descartes in 1638 and generalized by Euler to Euler's rule (Borho 1972).In order for such numbers to exist, there must be prime for two consecutive , leaving only the possibilities 1, 2, 3, 4, and 6, 7. Of these, is prime for , 4, and 7, giving the amicable pairs (220, 284), (17296, 18416), and (9363584, 9437056).In fact, various rules can be found that are analogous to Thâbit ibn Kurrah's. Denote a "Thâbit rule" by for given natural numbers and , a prime not dividing , , and polynomials . Then a necessary condition for the set of amicable pairs of the form (, 2) with..

A number is called an economical number if the number of digits in the prime factorization of (including powers) uses fewer digits than the number of digits in . The first few economical numbers are 125, 128, 243, 256, 343, 512, 625, 729, ... (OEIS A046759). Pinch shows that, under a plausible hypothesis related to the twin prime conjecture, there are arbitrarily long sequences of consecutive economical numbers, and exhibits such a sequence of length nine starting at 1034429177995381247.

A positive integer is th powerfree if there is no number such that ( divides ), i.e., there are no th powers or higher in the prime factorization of . A number which is free of all powers is therefore squarefree.

A superior highly composite number is a positive integer for which there is an such thatfor all , where the function counts the divisors of (Ramanujan 1962, pp. 87 and 115). It can be shown that all superior highly composite numbers are highly composite and that the th superior highly composite number has the form , where the factors are prime.The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the corresponding sequence of primes is 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ... (OEIS A000705).

A superabundant number is a composite number such that for all , where is the divisor function. Superabundant numbers are closely related to highly composite numbers, and the first 19 superabundant and highly composite numbers are the same. The first few are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, ... (OEIS A004394).

Given an integer , the Payam number is the smallest positive odd integer such that for every positive integer , the number is not divisible by any primes such that the multiplicative order of 2 is less than or equal to . Payam numbers are good candidates for searching for Proth primes, i.e., primes of the form , as well as primes of the form .The first few values of for , 3, ... are 3, 9, 15, 105, 105, 105, 105, 105, 165, 165, 75075, ... (OEIS A083556), and the first few values of are 3, 3, 45, 45, 45, 45, 45, 45, 45, 2145, ... (OEIS A083391).

An integer is called a super unitary perfect number ifwhere is the unitary divisor function. The first few are 2, 9, 165, 238, 1640, ... (OEIS A038843). It is not known if there exist any odd super unitary perfect numbers other than 9 and 165 (Yamada 2008).

The complexity of an integer is the least number of 1s needed to represent it using only additions, multiplications, and parentheses. For example, the numbers 1 through 10 can be minimally represented as(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)so the complexities for , 2, ..., are 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, ... (OEIS A005245).The smallest numbers of complexity , 2, ... are 1, 2, 3, 4, 5, 7, 10, 11, 17, 22, 23, 41, ... (OEIS A005520).

A Sierpiński number of the first kind is a number of the form . The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if is prime with , then must be of the form , making a Fermat number with . The first few of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127).The numbers of digits in the number is given bywhere is the ceiling function, so the numbers of digits in the first few candidates are 1, 3, 20, 617, 315653, 41373247568, ... (OEIS A089943).The only known prime Sierpiński numbers of the first kind are 2, 5, 257, with the first unknown case being . The status of Sierpiński numbers is summarized in the table below (Nielsen).status of 01prime ()13prime ()26composite with factor 311composite with factor 420composite with no factor known537composite with factor 670unknown7135unknown8264unknown9521unknown101034unknown112059composite with factor 124108unknown138205unknown1416398unknown1532783unknown1665552unknown17131089unknown..

A Mersenne number is a number of the form(1)where is an integer. The Mersenne numbers consist of all 1s in base-2, and are therefore binary repunits. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, ... (OEIS A000225), corresponding to , , , , ... in binary.The Mersenne numbers are also the numbers obtained by setting in a Fermat polynomial. They also correspond to Cunningham numbers .The number of digits in the Mersenne number is(2)where is the floor function, which, for large , gives(3)The number of digits in is the same as the number of digits in , namely 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, ... (OEIS A034887). The numbers of decimal digits in for , 1, ... are given by 1, 4, 31, 302, 3011, 30103, 301030, 3010300, 30103000, 301029996, ... (OEIS A114475), which correspond to the decimal expansion of (OEIS A007524).The numbers of prime factors of for , 2, ... are 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, ... (OEIS A046051), and the first few..

The hexanacci constant is the limiting ratio of adjacent hexanaccinumbers. It is the algebraic number(1)(2)(OEIS A118427), where denotes a polynomial root.

The heptanacci constant is the limiting ratio of adjacent heptanaccinumbers. It is the algebraic number(1)(2)(OEIS A118428), where denotes a polynomial root.

The tribonacci constant is ratio to which adjacent tribonaccinumbers tend, and is given by(1)(2)(3)(OEIS A058265).The tribonacci constant satisfies the identities(4)(5)(6)(7)(P. Moses, pers. comm., Feb. 21, 2005).The tribonacci constant is extremely prominent in the properties of the snubcube.

The tetranacci constant is ratio to which adjacent tetranaccinumbers tend, and is given by(1)(2)(OEIS A086088), where denotes a polynomial root.The tetranacci constant satisfies the identity(3)

The pentanacci constant is the limiting ratio of adjacent pentanaccinumbers. It is the algebraic number(1)(2)(OEIS A103814), where denotes a polynomial root.

An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..

A McNugget number is a positive integer that can be obtained by adding together orders of McDonald's® Chicken McNuggetsTM (prior to consuming any), which originally came in boxes of 6, 9, and 20 (Vardi 1991, pp. 19-20 and 233-234; Wah and Picciotto 1994, p. 186). All integers are McNugget numbers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. The value 43 therefore corresponds to the Frobenius number of .Since the Happy MealTM-sized nugget box (4 to a box) can now be purchased separately, the modern McNugget numbers are linear combinations of 4, 6, 9, and 20. These new-fangled numbers are much less interesting than before, with only 1, 2, 3, 5, 7, and 11 remaining as non-McNugget numbers. The value 11 therefore corresponds to the Frobenius number of .The greedy algorithm can be used to find a McNugget expansion of a given integer . This can also be done in the Wolfram Language using FrobeniusSolve[6,..

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers . Pick values of given by(1)where , 2, ... and is the floor function. We are then looking for factors such that(2)which means that only numbers with Legendre symbol (less than for trial divisor , where is the prime counting function) need be considered. The set of primes for which this is true is known as the factor base. Next, the congruences(3)must be solved for each in the factor base. Finally, a sieve is applied to find values of which can be factored completely using only the factor base. Gaussian elimination is then used as in Dixon's factorization method in order to find a product of the s, yielding a perfect square.The method requires about steps, improving on the continued fraction factorization algorithm by removing the 2 under the square root (Pomerance 1996). The use of multiple polynomials gives a better chance of factorization,..

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that be a prime is that be a prime of one of the forms , , ."Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff (1989) used the statement as an inspiration for what is now called the new Mersenne conjecture, which can be stated as follows.Consider an odd natural number . If two of the following conditions hold, then so does the third: 1. or , 2. is prime (a Mersenne prime), 3. is prime (a Wagstaff prime). This conjecture has been verified for all primes .Based on the distribution and heuristics of (cf. https://www.utm.edu/research/primes/mersenne/heuristic.html) the known Mersenne and Wagstaff prime exponents, it seems quite likely that there is only a finite number of exponents satisfying the criteria of the new Mersenne conjecture. In..

The first few numbers whose abundance absolute values are odd squares (excluding the trivial cases of powers of 2) are 98, 2116, 4232, 49928, 80656, 140450, 550564, 729632, ... (OEIS A188484).Kravitz conjectured that no numbers exist whose abundance is a (positive) odd square (Guy 2004). This conjecture is false with smallest counterexample(1)(2)and first few counterexamples given by 550564, 15038884, 57365476, ... (OEIS A188486).

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