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 th term describes the length of the th run Euler's totient function 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ... is the number of positive integers not greater than that are coprime with . Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... for , with and . Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... The prime numbers , with . A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. Partition numbers 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... The partition numbers, number of additive breakdowns of n. Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... for , with and . Sylvester's sequence 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ... for , with . Tribonacci numbers 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ... for , 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 cells. Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ... Bell numbers 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ... is the number of partitions of a set with elements. Euler zigzag numbers 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ... 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 cuts. Pell numbers 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ... for , with . Factorials 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... for , 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 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ... is the sum of divisors of a positive integer . Fermat numbers 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... for . 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, ... is equal to the sum of the proper divisors of . Ramanujan tau function 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ... Values of the Ramanujan tau function, at n = 1, 2, 3, ... Landau's function 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ... The largest order of permutation of 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 , with . Euclid–Mullin sequence 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ... is smallest prime factor of . 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, ... , 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 (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 , with . Sum of proper divisors 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ... is the sum of the proper divisors of the positive integer . 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 endpoints (and 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, ... is the number of times occurs, starting with . Perrin numbers 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ... for , 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 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ... , with . Primorials 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ... , the product of the first 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 for which there is an such that for all . Pronic numbers 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ... , with where 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 of the form for and . Ulam number 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... for is least number 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 such that if is coprime with . Woodall numbers 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ... , with . 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 such that for all ,
where denotes the sum-of-divisors function. 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 . Deficient numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ... Positive integers such that . Abundant numbers 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ... Positive integers such that . 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, ... Fortunate numbers 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ... The smallest integer such that is a prime number, where the primorial is the product of the first prime numbers. Semiperfect numbers 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ... A natural number 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 . 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, ... , i.e. product of first consecutive primes. Kaprekar numbers 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ... , where 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 . Radical of an integer 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... The radical of a positive integer is the product of the distinct prime numbers dividing . 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 where and are distinct primes congruent to . 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 for which Bernoulli numbers 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, ... -hyperperfect numbers, i.e. 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 for which consists only of composite numbers. Riesel numbers 509203, 762701, 777149, 790841, 992077, ... Odd for which consists only of composite numbers. Baum–Sweet sequence 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ... if the binary representation of contains no block of consecutive zeros of odd length; otherwise . Gijswijt's sequence 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ... The th term counts the maximal number of repeated blocks at the end of the subsequence from to Carol numbers −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ... Juggler sequence 0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ... If then else . Highly totient numbers 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ... Each number on this list has more solutions to the equation than any preceding . Euler numbers 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ... 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 such that there exists another number and Solution to Stepping Stone Puzzle 1, 16, 28, 38, 49, 60, ... The maximal value of the stepping stone puzzle
Figurate numbers