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List of integer sequences

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This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences.

General

Name	First elements	Short description	OEIS
Kolakoski sequence	1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...	The nth term describes the length of the nth run
Euler's totient function φ(n)	1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...	φ(n) is the number of positive integers not greater than n that are coprime with n.
Lucas numbers L(n)	2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...	for n ≥ 2, with and .
Prime numbers pn	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...	The prime numbers pn, with n ≥ 1. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
Partition numbers
Pn	1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...	The partition numbers, number of additive breakdowns of n.
Fibonacci numbers F(n)	0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...	for n ≥ 2, with and .
Sylvester's sequence	2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...	n ≥ 1}}, with .
Tribonacci numbers	0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...	for n ≥ 3, with .
Powers of 2	1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...	Powers of 2: 2n for n ≥ 0
Polyominoes	1, 1, 1, 2, 5, 12, 35, 108, 369, ...	The number of free polyominoes with n cells.
Catalan numbers Cn	1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...	C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} = \prod\limits_{k=2}^{n}\frac{n+k}{k},\quad n \ge 0.
Bell numbers Bn	1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...	Bn is the number of partitions of a set with n elements.
Euler zigzag numbers En	1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ...	En is the number of linear extensions of the "zig-zag" poset.
Lazy caterer's sequence	1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...	The maximal number of pieces formed when slicing a pancake with n cuts.
Pell numbers Pn	0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...	for n ≥ 2, with .
Factorials n!	1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...	n ≥ 1}}, with (empty product).
Derangements	1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...	Number of permutations of n elements with no fixed points.
Divisor function σ(n)	1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...	is the sum of divisors of a positive integer n.
Fermat numbers Fn	3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...	for n ≥ 0.
Polytrees	1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...	Number of oriented trees with n nodes.
Perfect numbers	6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...	n is equal to the sum of the proper divisors of n.
Ramanujan tau function	1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ...	Values of the Ramanujan tau function, τ(n) at n = 1, 2, 3, ...
Landau's function	1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...	The largest order of permutation of n elements.
Narayana's cows	1, 1, 1, 2, 3, 4, 6, 9, 13, 19, ...	The number of cows each year if each cow has one cow a year beginning its fourth year.
Padovan sequence	1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ...	for n ≥ 3, with .
Euclid–Mullin sequence	2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...	is smallest prime factor of a(1) a(2) ⋯ a(n) + 1.
Lucky numbers	1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...	A natural number in a set that is filtered by a sieve.
Prime powers	2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...	Positive integer powers of prime numbers
Central binomial coefficients	1, 2, 6, 20, 70, 252, 924, ...	{2n \choose n} = \frac{(2n)!}{(n!)^2}\text{ for all }n \geq 0, numbers in the center of even rows of Pascal's triangle
Motzkin numbers	1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...	The number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
Jordan–Pólya numbers	1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ...	Numbers that are the product of factorials.
Jacobsthal numbers	0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...	for n ≥ 2, with .
Sum of proper divisors s(n)	0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ...	is the sum of the proper divisors of the positive integer n.
Wedderburn–Etherington numbers	0, 1, 1, 1, 2, 3, 6, 11, 23, 46, ...	The number of binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n − 1 nodes in all).
Gould's sequence	1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, ...	Number of odd entries in row n of Pascal's triangle.
Semiprimes	4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...	Products of two primes, not necessarily distinct.
Golomb sequence	1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ...	a(n) is the number of times n occurs, starting with .
Perrin numbers Pn	3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ...	for n ≥ 3, with .
Sorting number	0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, ...	Used in the analysis of comparison sorts.
Cullen numbers Cn	1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...	, with n ≥ 0.
Primorials pn#	1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...	pn#, the product of the first n primes.
Highly composite numbers	1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...	A positive integer with more divisors than any smaller positive integer.
Superior highly composite numbers	2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...	A positive integer n for which there is an e 0 such that ≥ for all k 1.
Pronic numbers	0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...	, with n ≥ 0 where t(n) are the triangular numbers.
Markov numbers	1, 2, 5, 13, 29, 34, 89, 169, 194, ...	Positive integer solutions of .
Composite numbers	4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...	The numbers n of the form xy for x 1 and y 1.
Ulam number	1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...	for n 2, a(n) is least number a(n − 1) which is a unique sum of two distinct earlier terms; semiperfect.
Prime knots	0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...	The number of prime knots with n crossings.
Carmichael numbers	561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ...	Composite numbers n such that an − 1 ≡ 1 (mod n) if a is coprime with n.
Woodall numbers	1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...	n⋅2n − 1, with n ≥ 1.
Arithmetic numbers	1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, ...	An integer for which the average of its positive divisors is also an integer.
Colossally abundant numbers	2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...	A number n is colossally abundant if there is an ε 0 such that for all k 1,
Alcuin's sequence	0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, ...	Number of triangles with integer sides and perimeter n.
Deficient numbers	1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...	Positive integers n such that {{math	σ(n)
Abundant numbers	12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...	Positive integers n such that σ(n) 2n.
Untouchable numbers	2, 5, 52, 88, 96, 120, 124, 146, 162, 188, ...	Cannot be expressed as the sum of all the proper divisors of any positive integer.
Recamán's sequence	0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ...	"subtract if possible, otherwise add": a(0) = 0; for n 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
Look-and-say sequence	1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...	A = 'frequency' followed by 'digit'-indication.
Practical numbers	1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, ...	All smaller positive integers can be represented as sums of distinct factors of the number.
Alternating factorial	1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...	\sum_{k=0}^{n-1} (-1)^k (n-k)!
Fortunate numbers	3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ...	The smallest integer m 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
Semiperfect numbers	6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ...	A natural number n that is equal to the sum of all or some of its proper divisors.
Magic constants	15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, ...	Sum of numbers in any row, column, or diagonal of a magic square of order n ≥ 3.
Weird numbers	70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...	A natural number that is abundant but not semiperfect.
Farey sequence numerators	0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...
Farey sequence denominators	1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...
Euclid numbers	2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ...	pn# + 1, i.e. 1 + product of first n consecutive primes.
Kaprekar numbers	1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...	, where 0 n and .
Sphenic numbers	30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...	Products of 3 distinct primes.
Giuga numbers	30, 858, 1722, 66198, 2214408306, ...	Composite numbers so that for each of its distinct prime factors pi we have p_i^2 \,	\, (n - p_i).
Radical of an integer	1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...	The radical of a positive integer n is the product of the distinct prime numbers dividing n.
Thue–Morse sequence	0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...
Regular paperfolding sequence	1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...	At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence.
Blum integers	21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ...	Numbers of the form pq where p and q are distinct primes congruent to 3 (mod 4).
Magic numbers	2, 8, 20, 28, 50, 82, 126, ...	A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.
Superperfect numbers	2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...	Positive integers n for which
Bernoulli numbers Bn	1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...
Hyperperfect numbers	6, 21, 28, 301, 325, 496, 697, ...	k-hyperperfect numbers, i.e. n for which the equality holds.
Achilles numbers	72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...	Positive integers which are powerful but imperfect.
Primary pseudoperfect numbers	2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...	Satisfies a certain Egyptian fraction.
Erdős–Woods numbers	16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, ...	The length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints.
Sierpinski numbers	78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...	Odd k for which { k⋅2n + 1 : n ∈ \mathbb{N} } consists only of composite numbers.
Riesel numbers	509203, 762701, 777149, 790841, 992077, ...	Odd k for which { k⋅2n − 1 : n ∈ \mathbb{N} } consists only of composite numbers.
Baum–Sweet sequence	1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ...	if the binary representation of n contains no block of consecutive zeros of odd length; otherwise .
Gijswijt's sequence	1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ...	The nth term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n−1
Carol numbers	−1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...	a(n) = (2^n - 1)^2 - 2.
Juggler sequence	0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...	If n ≡ 0 (mod 2) then ⌊⌋ else ⌊n3/2⌋.
Highly totient numbers	1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...	Each number k on this list has more solutions to the equation than any preceding k.
Euler numbers	1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...	\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty \frac{E_n}{n!} \cdot t^n.
Polite numbers	3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...	A positive integer that can be written as the sum of two or more consecutive positive integers.
Erdős–Nicolas numbers	24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...	A number n such that there exists another number m and \sum_{d \mid n,\ d \leq m}\!d = n.
Solution to Stepping Stone Puzzle	1, 16, 28, 38, 49, 60, ...	The maximal value a(n) of the stepping stone puzzle

Figurate numbers

Main article: Figurate number

Name	First elements	Short description
Natural numbers	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...	The natural numbers (positive integers) n ∈ \mathbb{N}.
Triangular numbers t(n)	0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...	for n ≥ 1, with (empty sum).
Square numbers n2	0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...
Tetrahedral numbers T(n)	0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ...	T(n) is the sum of the first n triangular numbers, with (empty sum).
Square pyramidal numbers	0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ...	: The number of stacked spheres in a pyramid with a square base.
Cube numbers n3	0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...
Fifth powers	0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...
Star numbers	1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...	Sn = 6n(n − 1) + 1.
Stella octangula numbers	0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, ...	Stella octangula numbers: n(2n2 − 1), with n ≥ 0.

Types of primes

Main article: List of prime numbers

Name	First elements	Short description	OEIS
Mersenne prime exponents	2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...	Primes p such that 2p − 1 is prime.
Mersenne primes	3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ...	2p − 1 is prime, where p is a prime.
Wagstaff primes	3, 11, 43, 683, 2731, 43691, ...	A prime number p of the form p={{2^q+1}\over 3} where q is an odd prime.
Wieferich primes	1093, 3511		Primes p satisfying 2p−1 ≡ 1 (mod p2).
Sophie Germain primes	2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ...	A prime number p such that 2p + 1 is also prime.
Wilson primes	5, 13, 563		Primes p satisfying (p−1)! ≡ −1 (mod p2).
Happy numbers	1, 7, 10, 13, 19, 23, 28, 31, 32, 44, ...	The numbers whose trajectory under iteration of sum of squares of digits map includes 1.
Factorial primes	2, 3, 5, 7, 23, 719, 5039, 39916801, ...	A prime number that is one less or one more than a factorial (all factorials 1 are even).
Wolstenholme primes	16843, 2124679	Primes p satisfying {2p-1 \choose p-1} \equiv 1 \pmod{p^4}.
Ramanujan primes	2, 11, 17, 29, 41, 47, 59, 67, ...	The nth Ramanujan prime is the least integer Rn for which π(x) − π(x/2) ≥ n, for all x ≥ Rn.

Base-dependent

Main article: Base-dependent integer sequences

Name	First elements	Short description
Aronson's sequence	1, 4, 11, 16, 24, 29, 33, 35, 39, 45, ...	"t" is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas.
Palindromic numbers	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ...	A number that remains the same when its digits are reversed.
Permutable primes	2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ...	The numbers for which every permutation of digits is a prime.
Harshad numbers in base 10	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ...	A Harshad number in base 10 is an integer that is divisible by the sum of its digits (when written in base 10).
Factorions	1, 2, 145, 40585, ...	A natural number that equals the sum of the factorials of its decimal digits.
Circular primes	2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ...	The numbers which remain prime under cyclic shifts of digits.
Home prime	1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ...	For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; if no prime is ever reached.
Undulating numbers	101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ...	A number that has the digit form ababab.
Equidigital numbers	1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, ...	A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1.
Extravagant numbers	4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ...	A number that has fewer digits than the number of digits in its prime factorization (including exponents).
Pandigital numbers	1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, ...	Numbers containing the digits 0–9 such that each digit appears exactly once.

References

OEIS core sequences

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