BREAKING NEWS
List of integer sequences

## Summary

This is a list of notable integer sequences.

## General

OEIS link Name First elements Short description
A000002 Kolakoski sequence {1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...} The nth term describes the length of the nth run
A000010 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.
A000032 Lucas numbers L(n) {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...} L(n) = L(n − 1) + L(n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1.
A000040 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.
A000041 Partition numbers
Pn
{1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...} The partition numbers, number of additive breakdowns of n.
A000045 Fibonacci numbers F(n) {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...} F(n) = F(n − 1) + F(n − 2) for n ≥ 2, with F(0) = 0 and F(1) = 1.
A000058 Sylvester's sequence {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...} a(n + 1) = a(n)⋅a(n − 1)⋅ ⋯ ⋅a(0) + 1 = a(n)2a(n) + 1 for n ≥ 1, with a(0) = 2.
A000073 Tribonacci numbers {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...} T(n) = T(n − 1) + T(n − 2) + T(n − 3) for n ≥ 3, with T(0) = 0 and T(1) = T(2) = 1.
A000079 Powers of 2 {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...} Powers of 2: 2n for n ≥ 0
A000105 Polyominoes {1, 1, 1, 2, 5, 12, 35, 108, 369, ...} The number of free polyominoes with n cells.
A000108 Catalan numbers Cn {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...} ${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}},\quad n\geq 0.}$
A000110 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.
A000111 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.
A000124 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.
A000129 Pell numbers Pn {0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...} a(n) = 2a(n − 1) + a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A000142 Factorials n! {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...} n! = 1⋅2⋅3⋅4⋅ ⋯ ⋅n for n ≥ 1, with 0! = 1 (empty product).
A000166 Derangements {1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...} Number of permutations of n elements with no fixed points.
A000203 Divisor function σ(n) {1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...} σ(n) := σ1(n) is the sum of divisors of a positive integer n.
A000215 Fermat numbers Fn {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...} Fn = 22n + 1 for n ≥ 0.
A000238 Polytrees {1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...} Number of oriented trees with n nodes.
A000396 Perfect numbers {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...} n is equal to the sum s(n) = σ(n) − n of the proper divisors of n.
A000594 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, ...
A000793 Landau's function {1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...} The largest order of permutation of n elements.
A000930 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.
A000931 Padovan sequence {1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ...} P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = P(1) = P(2) = 1.
A000945 Euclid–Mullin sequence {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...} a(1) = 2; a(n + 1) is smallest prime factor of a(1) a(2) ⋯ a(n) + 1.
A000959 Lucky numbers {1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...} A natural number in a set that is filtered by a sieve.
A000961 Prime powers {1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...} Positive integer powers of prime numbers
A000984 Central binomial coefficients {1, 2, 6, 20, 70, 252, 924, ...} ${\displaystyle {2n \choose n}={\frac {(2n)!}{(n!)^{2}}}{\text{ for all }}n\geq 0}$ , numbers in the center of even rows of Pascal's triangle
A001006 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.
A001013 Jordan–Pólya numbers {1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64. ...} Numbers that are the product of factorials.
A001045 Jacobsthal numbers {0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...} a(n) = a(n − 1) + 2a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A001065 Sum of proper divisors s(n) {0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ...} s(n) = σ(n) − n is the sum of the proper divisors of the positive integer n.
A001190 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).
A001316 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.
A001358 Semiprimes {4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...} Products of two primes, not necessarily distinct.
A001462 Golomb sequence {1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ...} a(n) is the number of times n occurs, starting with a(1) = 1.
A001608 Perrin numbers Pn {3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ...} P(n) = P(n−2) + P(n−3) for n ≥ 3, with P(0) = 3, P(1) = 0, P(2) = 2.
A001855 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.
A002064 Cullen numbers Cn {1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...} Cn = n⋅2n + 1, with n ≥ 0.
A002110 Primorials pn# {1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...} pn#, the product of the first n primes.
A002182 Highly composite numbers {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...} A positive integer with more divisors than any smaller positive integer.
A002201 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 d(n)/ned(k)/ke for all k > 1.
A002378 Pronic numbers {0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...} a(n) = 2t(n) = n(n + 1), with n ≥ 0 where t(n) are the triangular numbers.
A002559 Markov numbers {1, 2, 5, 13, 29, 34, 89, 169, 194, ...} Positive integer solutions of x2 + y2 + z2 = 3xyz.
A002808 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.
A002858 Ulam number {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...} a(1) = 1; a(2) = 2; for n > 2, a(n) is least number > a(n − 1) which is a unique sum of two distinct earlier terms; semiperfect.
A002863 Prime knots {0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...} The number of prime knots with n crossings.
A002997 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.
A003261 Woodall numbers {1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...} n⋅2n − 1, with n ≥ 1.
A003601 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.
A004490 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,
${\displaystyle {\frac {\sigma (n)}{n^{1+\varepsilon }}}\geq {\frac {\sigma (k)}{k^{1+\varepsilon }}},}$

where σ denotes the sum-of-divisors function.

A005044 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.
A005100 Deficient numbers {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...} Positive integers n such that σ(n) < 2n.
A005101 Abundant numbers {12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...} Positive integers n such that σ(n) > 2n.
A005114 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.
A005132 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.
A005150 Look-and-say sequence {1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...} A = 'frequency' followed by 'digit'-indication.
A005153 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.
A005165 Alternating factorial {1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...} n! − (n−1)! + (n−2)! − ... 1!.
A005235 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.
A005835 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.
A006003 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.
A006037 Weird numbers {70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...} A natural number that is abundant but not semiperfect.
A006842 Farey sequence numerators {0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...}
A006843 Farey sequence denominators {1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...}
A006862 Euclid numbers {2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ...} pn# + 1, i.e. 1 + product of first n consecutive primes.
A006886 Kaprekar numbers {1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...} X2 = Abn + B, where 0 < B < bn and X = A + B.
A007304 Sphenic numbers {30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...} Products of 3 distinct primes.
A007850 Giuga numbers {30, 858, 1722, 66198, 2214408306, ...} Composite numbers so that for each of its distinct prime factors pi we have ${\displaystyle p_{i}^{2}\,|\,(n-p_{i})}$ .
A007947 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.
A010060 Thue–Morse sequence {0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...}
A014577 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.
A016105 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).
A018226 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.
A019279 Superperfect numbers {2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...} Positive integers n for which σ2(n) = σ(σ(n)) = 2n.
A027641 Bernoulli numbers Bn {1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...}
A034897 Hyperperfect numbers {6, 21, 28, 301, 325, 496, 697, ...} k-hyperperfect numbers, i.e. n for which the equality n = 1 + k (σ(n) − n − 1) holds.
A052486 Achilles numbers {72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...} Positive integers which are powerful but imperfect.
A054377 Primary pseudoperfect numbers {2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...} Satisfies a certain Egyptian fraction.
A059756 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.
A076336 Sierpinski numbers {78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...} Odd k for which { k⋅2n + 1 : n${\displaystyle \mathbb {N} }$  } consists only of composite numbers.
A076337 Riesel numbers {509203, 762701, 777149, 790841, 992077, ...} Odd k for which { k⋅2n − 1 : n${\displaystyle \mathbb {N} }$  } consists only of composite numbers.
A086747 Baum–Sweet sequence {1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ...} a(n) = 1 if the binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0.
A090822 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
A093112 Carol numbers {−1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...} ${\displaystyle a(n)=(2^{n}-1)^{2}-2.}$
A094683 Juggler sequence {0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...} If n ≡ 0 (mod 2) then n else n3/2.
A097942 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 φ(x) = k than any preceding k.
A122045 Euler numbers {1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...} ${\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.}$
A138591 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.
A194472 Erdős–Nicolas numbers {24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...} A number n such that there exists another number m and ${\displaystyle \sum _{d\mid n,\ d\leq m}\!d=n.}$
A337663 Solution to Stepping Stone Puzzle {1, 16, 28, 38, 49, 60 ...} The maximal value a(n) of the stepping stone puzzle

## Figurate numbers

OEIS link Name First elements Short description
A000027 Natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...} The natural numbers (positive integers) n${\displaystyle \mathbb {N} }$ .
A000217 Triangular numbers t(n) {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...} t(n) = C(n + 1, 2) = n(n + 1)/2 = 1 + 2 + ⋯ + n for n ≥ 1, with t(0) = 0 (empty sum).
A000290 Square numbers n2 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...} n2 = n × n
A000292 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 T(0) = 0 (empty sum).
A000330 Square pyramidal numbers {0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ...} n(n + 1)(2n + 1)/6 : The number of stacked spheres in a pyramid with a square base.
A000578 Cube numbers n3 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...} n3 = n × n × n
A000584 Fifth powers {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...} n5
A003154 Star numbers {1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...} Sn = 6n(n − 1) + 1.
A007588 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

OEIS link Name First elements Short description
A000043 Mersenne prime exponents {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...} Primes p such that 2p − 1 is prime.
A000668 Mersenne primes {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ...} 2p − 1 is prime, where p is a prime.
A000979 Wagstaff primes {3, 11, 43, 683, 2731, 43691, ...} A prime number p of the form ${\displaystyle p={{2^{q}+1} \over 3}}$  where q is an odd prime.
A001220 Wieferich primes {1093, 3511} Primes ${\displaystyle p}$  satisfying 2p−1 ≡ 1 (mod p2).
A005384 Sophie Germain primes {2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ...} A prime number p such that 2p + 1 is also prime.
A007540 Wilson primes {5, 13, 563} Primes ${\displaystyle p}$  satisfying (p−1)! ≡ −1 (mod p2).
A007770 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.
A088054 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).
A088164 Wolstenholme primes {16843, 2124679} Primes ${\displaystyle p}$  satisfying ${\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}}}$ .
A104272 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 xRn.

## Base-dependent

OEIS link Name First elements Short description
A005224 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.
A002113 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.
A003459 Permutable primes {2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ...} The numbers for which every permutation of digits is a prime.
A005349 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).
A014080 Factorions {1, 2, 145, 40585, ...} A natural number that equals the sum of the factorials of its decimal digits.
A016114 Circular primes {2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ...} The numbers which remain prime under cyclic shifts of digits.
A037274 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; a(n) = −1 if no prime is ever reached.
A046075 Undulating numbers {101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ...} A number that has the digit form ababab.
A046758 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.
A046760 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).
A050278 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