# Category: Integer sequences

Perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another in
Factorial prime
A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (se
Magic number (physics)
In nuclear physics, a magic number is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the atomic nucleus. As a result, atomic nucl
Ulam number
In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with U1
Spt function
The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each partition of a positive integer. It is related to the partition fu
Pillai sequence
The Pillai sequence is the sequence of integers that have a record number of terms in their greedy representations as sums of prime numbers (and one).It is named after Subbayya Sivasankaranarayana Pil
Descartes number
In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the
Semiperfect number
In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of al
Sociable number
In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable seque
Seventh power
In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n. Seventh powers are also formed by multiplying
Kolakoski sequence
In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths in its own run-length e
Somos sequence
In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their definin
Smooth number
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number whose every prime factor is at most 7,
Pronic number
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form The study of these numbers dates back to Aristotle. They are also called oblong numbers, hete
Fermi–Dirac prime
In number theory, a Fermi–Dirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to Fermi–Dirac statistics in physics based on the fact that each intege
Lobb number
In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced p
Interprime
In mathematics, an interprime is the average of two consecutive odd primes. For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are: 4, 6, 9, 12, 15, 18, 21, 2
Sixth power
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by
Stanley sequence
In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions. If is a finite set of non-negative integer
Wedderburn–Etherington number
The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few number
K-regular sequence
In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regul
Refactorable number
A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that . The first few refactorable numbers are listed in (seque
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of
Dedekind number
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) counts the number of monotone boolean f
Superabundant number
In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number n is called superabundant precisely when, for all m < n where σ denotes the s
Arithmetic number
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is w
Power of two
A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are c
Schröder number
In mathematics, the Schröder number also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner of an grid to the northeast corner using
Doubly triangular number
In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if denotes the th triangula
Amicable triple
In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers. In another equivalent charac
Carmichael number
In number theory, a Carmichael number is a composite number , which in modular arithmetic satisfies the congruence relation: for all integers which are relatively prime to . The relation may also be e
3x + 1 semigroup
In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the seque
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively
Eighth power
In arithmetic and algebra the eighth power of a number n is the result of multiplying eight instances of n together. So: n8 = n × n × n × n × n × n × n × n. Eighth powers are also formed by multiplyin
Highly totient number
A highly totient number is an integer that has more solutions to the equation , where is Euler's totient function, than any integer below it. The first few highly totient numbers are 1, 2, 4, 8, 12, 2
Meander (mathematics)
In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number
Multiplicative partition
In number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they diffe
Calkin–Wilf tree
In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers. The tree is rooted at the number 1, and any rational number expressed in
Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the fi
Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci num
Beatty sequence
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are
Fractal sequence
In mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ... If the first occurrence of each
Irrationality sequence
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series exists (that is, it
Hemiperfect number
In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the divisor function, the sum of all
Large set (combinatorics)
In combinatorial mathematics, a large set of positive integers is one such that the infinite sum of the reciprocals diverges. A small set is any subset of the positive integers that is not large; that
Primary pseudoperfect number
In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation where the sum is over only the prime divisors of N.
Constant-recursive sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or mor
Superior highly composite number
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to some positive power of itself than any other number. It is a stronger re
Lunar arithmetic
Lunar arithmetic, formerly called dismal arithmetic, is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations. Thus, in lunar
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For
Squared triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, The same equation may be written more compactly using the mathematical notation for summation: This
Ménage problem
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so t
Wild number
Originally, wild numbers are the numbers supposed to belong to a fictional sequence of numbers imagined to exist in the mathematical world of the mathematical fiction The Wild Numbers authored by Phil
Noncototient
In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where
Giuga number
A Giuga number is a composite number n such that for each of its distinct prime factors pi we have , or equivalently such that for each of its distinct prime factors pi we have . The Giuga numbers are
Göbel's sequence
In mathematics, Göbel's sequence is a sequence of rational numbers defined by the recurrence relation with starting value Göbel's sequence starts with 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... (s
Lucky numbers of Euler
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot be prime since n2
Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also descri
Square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime
Perfect totient number
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totien
Elliptic divisibility sequence
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined
Highly powerful number
In elementary number theory, a highly powerful number is a positive integer that satisfies a property introduced by the Indo-Canadian mathematician Mathukumalli V. Subbarao. The set of highly powerful
Partition function (number theory)
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 +
Lah number
In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the th derivatives of .
Greedy algorithm for Egyptian fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a re
Lucas sequence
In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation where and are fixed integers. Any sequence satisfying this recurrence rela
Primitive semiperfect number
No description available.
Kostka number
In mathematics, the Kostka number Kλμ (depending on two integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They w
Divisor sum identities
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or
Hyperfactorial
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to .
Jacobsthal number
In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence for
Singly and doubly even
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names ar
Weird number
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater
Lucas chain
In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician Édouard Lucas. It is a sequence a0, a1, a2, a3, ... that satisfies a0=1, and for each k > 0: ak
Cube (algebra)
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together.The cube of a number or any other mathematical expression is deno
Fortunate number
A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n p
Composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself.
Polite number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impo
Golomb sequence
In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a monotonically increasing integer sequence where an is the number of times that n occurs
Leyland number
In number theory, a Leyland number is a number of the form where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are 8, 17, 32,
Strobogrammatic number
A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. In other words, the numeral looks the same right-side up and upside
Ordered Bell number
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing
Sorting number
In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-cas
Mian–Chowla sequence
In mathematics, the Mian–Chowla sequence is an integer sequence definedrecursively in the following way. The sequence starts with Then for , is the smallest integer such that every pairwise sum is dis
Quasiperfect number
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors
Recamán's sequence
In mathematics and computer science, Recamán's sequence is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, t
Boustrophedon transform
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular a
Highly cototient number
In number theory, a branch of mathematics, a highly cototient number is a positive integer which is above 1 and has more solutions to the equation than any other integer below and above 1. Here, is Eu
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17,
Amicable numbers
Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(a)=b and σ(b)=a, where σ(n) is equal to th
Double factorial
In mathematics, the double factorial or semifactorial of a number n, denoted by n‼, is the product of all the integers from 1 up to n that have the same parity (odd or even) as n. That is, For even n,
Colossally abundant number
In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Formally, a number n is said to be colossally a
Znám's problem
In number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's proble
Almost perfect number
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function
Congruum
In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares.That is, if , , and (for integers , , and ) are three sq
Superperfect number
In mathematics, a superperfect number is a positive integer n that satisfies where σ is the divisor summatory function. Superperfect numbers are a generalization of perfect numbers. The term was coine
Deficient number
In number theory, a deficient number or defective number is a number n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot
Dirichlet series inversion
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little kno
In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values and the recurrence relation The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16,
Euler numbers
In mathematics, the Euler numbers are a sequence En of integers (sequence in the OEIS) defined by the Taylor series expansion , where is the hyperbolic cosine function. The Euler numbers are related t
Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the Frenc
Lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that
Rough number
A k-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to k. k-roughness has alternately been defined as requiring all prime fa
Sequences (book)
Sequences is a mathematical monograph on integer sequences. It was written by Heini Halberstam and Klaus Roth, published in 1966 by the Clarendon Press, and republished in 1983 with minor corrections
Highly abundant number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Hi
List of integer sequences
This is a list of notable integer sequences.
Odious number
In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. In computer science, an odious number is said to have odd parity.
Zeisel number
A Zeisel number, named after , is a square-free integer k with at least three prime factors which fall into the pattern where a and b are some integer constants and x is the index number of each prime
Double Mersenne number
In mathematics, a double Mersenne number is a Mersenne number of the form where p is prime.
Eulerian number
In combinatorics, the Eulerian number A(n, m) is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). The
Lucas–Carmichael number
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that 1. * if p is a prime factor of n, then p + 1 is a factor of n + 1; 2. * n is odd and square-free. The first cond
Loeschian number
In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called theLoeschian numbers. These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They
Hermite number
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
Achilles number
An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime factor p of n, p2 is also a divisor. In other words, every prime
Lehmer sequence
In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.
Euclid–Mullin sequence
The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integ
Wolstenholme number
A Wolstenholme number is a number that is the numerator of the generalized harmonic number Hn,2. The first such numbers are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (sequence in the OEIS). Thes
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This se
Alternating factorial
In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers. This is the same as their sum, with the odd-indexed factorials mul
Highly composite number
A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at leas
Primitive abundant number
In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: 1. * The sum of its proper di
Betrothed numbers
Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (m, n) a
Friendly number
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "ab
Integer sequence prime
In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to
Mersenne number
No description available.
Complete sequence
In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, t
Hilbert number
In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 ). The Hilbert numbers were named after David Hilbert.The sequence of Hilbert numbers begins 1, 5,
Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n4 = n × n × n × n Fourth powers are also formed by multiplying a number by its
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intelle
Bell number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Jap
Primefree sequence
In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fi
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 18
Motzkin number
In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin nu
Lambek–Moser theorem
In combinatorial number theory, the Lambek–Moser theorem splits the natural numbers into two complementary sets using any non-decreasing function and its inverse. It extends Rayleigh's theorem, which
Euclid number
In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematicia
Prime omega function
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby (little omega) counts each distinct prime factor, whereas the related function (big omega)
Multiply perfect number
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (
Moser–de Bruijn sequence
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4, or equivalently the numbers w
Stirling number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus
Integer complexity
In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is a
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relation P(n) = P(n − 2) + P(n − 3) for n > 2, with initial values P(0) = 3, P(1) = 0, P(2) = 2. The sequence of Perrin numbers starts
Schröder–Hipparchus number
In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses i
Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a r
Coordination sequence
In crystallography and the theory of infinite vertex-transitive graphs, the coordination sequence of a vertex is an integer sequence that counts how many vertices are at each possible distance from .
Genocchi number
In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation The first few Genocchi numbers are 0, −1, −1, 0, 1, 0, −3, 0, 17 (sequence in
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and
Sparsely totient number
In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n, where is Euler's totient function. The first few sparsely toti
Blum integer
In mathematics, a natural number n is a Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t + 3, for som
Jordan–Pólya number
In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, is a Jordan–Pól
Harmonic divisor number
In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few h
Sphenic number
In number theory, a sphenic number (from Ancient Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, ther
Derangement
In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no
Woodall number
In number theory, a Woodall number (Wn) is any natural number of the form for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence in the OEIS).
Centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center
Fifth power (algebra)
In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together: n5 = n × n × n × n × n. Fifth powers are also formed by multiplying a nu
Primorial prime
In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes). Primality tests show that pn# − 1 is prime for n = 2
Cullen number
In mathematics, a Cullen number is a member of the integer sequence (where is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbe
Unusual number
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than . A k-smooth number has all its prime factors less than or equal to k, therefore, an unusu
Leonardo number
The Leonardo numbers are a sequence of numbers given by the recurrence: Edsger W. Dijkstra used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail. A Leonardo
Størmer number
In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer for which the greatest prime factor of is greater than or equal to . They are named after Carl Størmer.
Triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n
Ergodic sequence
In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.
Evil number
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and fo
Gregory coefficients
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin s
Exponential factorial
The exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on and so forth in a right-grouping manner. That is, The exponenti
Home prime
In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth
Idoneal number
In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is rela
Wilson quotient
The Wilson quotient W(p) is defined as: If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is ca
Poly-Bernoulli number
In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as where Li is the polylogarithm. The are the usual Bernoulli numbers. Moreover, the Generalization of Poly-Bernoulli num
Untouchable number
An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a
Automatic sequence
In mathematics and theoretical computer science, an automatic sequence (also called a k-automatic sequence or a k-recognizable sequence when one wants to indicate that the base of the numerals used is
Hyperperfect number
In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A
Katydid sequence
The Katydid sequence is a sequence of numbers first defined in Clifford A. Pickover's book Wonders of Numbers (2001).
Equidigital number
In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, inclu
Juggler sequence
In number theory, a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation:
Nontotient
In number theory, a nontotient is a positive integer n which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(x) = n has no solution x. In other w
Erdős–Nicolas number
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors.That is, a number n is Erdős–Nicolas number when there exists another
Regular number
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 =
Abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper diviso
Journal of Integer Sequences
The Journal of Integer Sequences is a peer-reviewed open-access academic journal in mathematics, specializing in research papers about integer sequences. It was founded in 1998 by Neil Sloane. Sloane
Power of 10
A power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one i
Weak ordering
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders a
Sum-free sequence
In mathematics, a sum-free sequence is an increasing sequence of positive integers, such that no term can be represented as a sum of any subset of the preceding elements of the same sequence. This dif
Sublime number
In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number. The number 12,
Ban number
In recreational mathematics, a ban number is a number that does not contain a particular letter when spelled out in English; in other words, the letter is "banned." Ban numbers are not precisely defin
Super-Poulet number
A super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor d divides 2d − 2. For example, 341 is a super-Poulet number: it has positive divisors {1, 11, 31, 341} and we ha
Zimmert set
In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (p
Behrend sequence
In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend.
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne,
Divisibility sequence
In mathematics, a divisibility sequence is an integer sequence indexed by positive integers n such that for all m, n. That is, whenever one index is a multiple of another one, then the corresponding t
Powerful number
A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m o
Almost prime
In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in
Unitary perfect number
A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself (a divisor d of a number n is a unitary divisor if d and n/d share no c
Lucky number
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates n
Erdős–Woods number
In number theory, a positive integer k is said to be an Erdős–Woods number if it has the following property:there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecut
Narayana number
In combinatorics, the Narayana numbers form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T.
Thabit number
In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form for a non-negative integer n. The first few Thabit numbers are: 2, 5, 11, 23, 47, 95, 191, 383, 767,
Baire space (set theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements a
Superfactorial
In mathematics, and more specifically number theory, the superfactorial of a positive integer is the product of the first factorials. They are a special case of the Jordan–Pólya numbers, which are pro
Semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime num
Primitive permutation group
In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into
Delannoy number
In mathematics, a Delannoy number describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east
Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, sinc
Nonhypotenuse number
In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenus
Alcuin's sequence
In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: The sequence begins with these integers: 0, 0, 0, 1, 0, 1, 1, 2, 1, 3,
Goodstein's theorem
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby an
Gould's sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2,
Practical number
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a pract