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 Template:Mathth term describes the length of the Template:Mathth run Template:OEIS link
Euler's totient function Template:Math 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ... Template:Math is the number of positive integers not greater than Template:Math that are coprime with Template:Math. Template:OEIS link
Lucas numbers Template:Math 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... Template:Math for Template:Math, with Template:Math and Template:Math. Template:OEIS link
Prime numbers Template:Math 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... The prime numbers Template:Math, with Template:Math. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. Template:OEIS link
Partition numbers
Template:Math
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... The partition numbers, number of additive breakdowns of n. Template:OEIS link
Fibonacci numbers Template:Math 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Template:Math for Template:Math, with Template:Math and Template:Math. Template:OEIS link
Sylvester's sequence 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ... a(n+1)=k=0na(k)+1=a(n)2a(n)+1 for Template:Math, with Template:Math. Template:OEIS link
Tribonacci numbers 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ... Template:Math for Template:Math, with Template:Math. Template:OEIS link
Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... Powers of 2: 2n for n ≥ 0 Template:OEIS link
Polyominoes 1, 1, 1, 2, 5, 12, 35, 108, 369, ... The number of free polyominoes with Template:Math cells. Template:OEIS link
Catalan numbers Template:Math 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ... Cn=1n+1(2nn)=(2n)!(n+1)!n!=k=2nn+kk,n0. Template:OEIS link
Bell numbers Template:Math 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ... Template:Math is the number of partitions of a set with Template:Math elements. Template:OEIS link
Euler zigzag numbers Template:Math 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ... Template:Math is the number of linear extensions of the "zig-zag" poset. Template:OEIS link
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 Template:Math cuts. Template:OEIS link
Pell numbers Template:Math 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ... Template:Math for Template:Math, with Template:Math. Template:OEIS link
Factorials Template:Math 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... n!=k=1nk for Template:Math, with Template:Math (empty product). Template:OEIS link
Derangements 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ... Number of permutations of n elements with no fixed points. Template:OEIS link
Divisor function Template:Math 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ... Template:Math is the sum of divisors of a positive integer Template:Math. Template:OEIS link
Fermat numbers Template:Math 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... Template:Math for Template:Math. Template:OEIS link
Polytrees 1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ... Number of oriented trees with n nodes. Template:OEIS link
Perfect numbers 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ... Template:Math is equal to the sum Template:Math of the proper divisors of Template:Math. Template:OEIS link
Ramanujan tau function 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ... Values of the Ramanujan tau function, Template:Math at n = 1, 2, 3, ... Template:OEIS link
Landau's function 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ... The largest order of permutation of Template:Math elements. Template:OEIS link
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. Template:OEIS link
Padovan sequence 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ... Template:Math for Template:Math, with Template:Math. Template:OEIS link
Euclid–Mullin sequence 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ... Template:Math is smallest prime factor of Template:Math. Template:OEIS link
Lucky numbers 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ... A natural number in a set that is filtered by a sieve. Template:OEIS link
Prime powers 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ... Positive integer powers of prime numbers Template:OEIS link
Central binomial coefficients 1, 2, 6, 20, 70, 252, 924, ... (2nn)=(2n)!(n!)2 for all n0, numbers in the center of even rows of Pascal's triangle Template:OEIS link
Motzkin numbers 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ... The number of ways of drawing any number of nonintersecting chords joining Template:Math (labeled) points on a circle. Template:OEIS link
Jordan–Pólya numbers 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ... Numbers that are the product of factorials. Template:OEIS link
Jacobsthal numbers 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... Template:Math for Template:Math, with Template:Math. Template:OEIS link
Sum of proper divisors Template:Math 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ... Template:Math is the sum of the proper divisors of the positive integer Template:Math. Template:OEIS link
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 Template:Math endpoints (and Template:Math nodes in all). Template:OEIS link
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. Template:OEIS link
Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... Products of two primes, not necessarily distinct. Template:OEIS link
Golomb sequence 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ... Template:Math is the number of times Template:Math occurs, starting with Template:Math. Template:OEIS link
Perrin numbers Template:Math 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ... Template:Math for Template:Math, with Template:Math. Template:OEIS link
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. Template:OEIS link
Cullen numbers Template:Math 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ... Template:Math, with Template:Math. Template:OEIS link
Primorials Template:Math 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ... Template:Math, the product of the first Template:Math primes. Template:OEIS link
Highly composite numbers 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... A positive integer with more divisors than any smaller positive integer. Template:OEIS link
Superior highly composite numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ... A positive integer Template:Math for which there is an Template:Math such that Template:Math for all Template:Math. Template:OEIS link
Pronic numbers 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ... Template:Math, with Template:Math where Template:Math are the triangular numbers. Template:OEIS link
Markov numbers 1, 2, 5, 13, 29, 34, 89, 169, 194, ... Positive integer solutions of Template:Math. Template:OEIS link
Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... The numbers Template:Math of the form Template:Math for Template:Math and Template:Math. Template:OEIS link
Ulam number 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... Template:Math for Template:Math is least number Template:Math which is a unique sum of two distinct earlier terms; semiperfect. Template:OEIS link
Prime knots 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ... The number of prime knots with n crossings. Template:OEIS link
Carmichael numbers 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ... Composite numbers Template:Math such that Template:Math if Template:Math is coprime with Template:Math. Template:OEIS link
Woodall numbers 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ... Template:Math, with Template:Math. Template:OEIS link
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. Template:OEIS link
Colossally abundant numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ... A number n is colossally abundant if there is an Template:Math such that for all Template:Math,
σ(n)n1+εσ(k)k1+ε,

where Template:Mvar denotes the sum-of-divisors function.

Template:OEIS link
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 Template:Math. Template:OEIS link
Deficient numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ... Positive integers Template:Math such that Template:Math. Template:OEIS link
Abundant numbers 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ... Positive integers Template:Math such that Template:Math. Template:OEIS link
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. Template:OEIS link
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. Template:OEIS link
Look-and-say sequence 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ... A = 'frequency' followed by 'digit'-indication. Template:OEIS link
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. Template:OEIS link
Alternating factorial 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ... k=0n1(1)k(nk)! Template:OEIS link
Fortunate numbers 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ... The smallest integer Template:Math such that Template:Math is a prime number, where the primorial Template:Math is the product of the first Template:Math prime numbers. Template:OEIS link
Semiperfect numbers 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ... A natural number Template:Math that is equal to the sum of all or some of its proper divisors. Template:OEIS link
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 Template:Math. Template:OEIS link
Weird numbers 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ... A natural number that is abundant but not semiperfect. Template:OEIS link
Farey sequence numerators 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...   Template:OEIS link
Farey sequence denominators 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...   Template:OEIS link
Euclid numbers 2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ... Template:Math, i.e. Template:Math product of first Template:Math consecutive primes. Template:OEIS link
Kaprekar numbers 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ... Template:Math, where Template:Math and Template:Math. Template:OEIS link
Sphenic numbers 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ... Products of 3 distinct primes. Template:OEIS link
Giuga numbers 30, 858, 1722, 66198, 2214408306, ... Composite numbers so that for each of its distinct prime factors pi we have pi2|(npi). Template:OEIS link
Radical of an integer 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... The radical of a positive integer Template:Math is the product of the distinct prime numbers dividing Template:Math. Template:OEIS link
Thue–Morse sequence 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ... Template:OEIS link
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. Template:OEIS link
Blum integers 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ... Numbers of the form Template:Math where Template:Mvar and Template:Mvar are distinct primes congruent to Template:Math. Template:OEIS link
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. Template:OEIS link
Superperfect numbers 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ... Positive integers Template:Math for which Template:Math Template:OEIS link
Bernoulli numbers Template:Math 1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...   Template:OEIS link
Hyperperfect numbers 6, 21, 28, 301, 325, 496, 697, ... Template:Math-hyperperfect numbers, i.e. Template:Math for which the equality Template:Math holds. Template:OEIS link
Achilles numbers 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ... Positive integers which are powerful but imperfect. Template:OEIS link
Primary pseudoperfect numbers 2, 6, 42, 1806, 47058, 2214502422, 52495396602, ... Satisfies a certain Egyptian fraction. Template:OEIS link
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. Template:OEIS link
Sierpinski numbers 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ... Odd Template:Math for which Template:Math consists only of composite numbers. Template:OEIS link
Riesel numbers 509203, 762701, 777149, 790841, 992077, ... Odd Template:Math for which Template:Math consists only of composite numbers. Template:OEIS link
Baum–Sweet sequence 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ... Template:Math if the binary representation of Template:Math contains no block of consecutive zeros of odd length; otherwise Template:Math. Template:OEIS link
Gijswijt's sequence 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ... The Template:Mathth term counts the maximal number of repeated blocks at the end of the subsequence from Template:Math to Template:Math Template:OEIS link
Carol numbers −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ... a(n)=(2n1)22. Template:OEIS link
Juggler sequence 0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ... If Template:Math then Template:Math else Template:Math. Template:OEIS link
Highly totient numbers 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ... Each number Template:Mvar on this list has more solutions to the equation Template:Math than any preceding Template:Math. Template:OEIS link
Euler numbers 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ... 1cosht=2et+et=n=0Enn!tn. Template:OEIS link
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. Template:OEIS link
Erdős–Nicolas numbers 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ... A number Template:Mvar such that there exists another number Template:Mvar and dn, dmd=n. Template:OEIS link
Solution to Stepping Stone Puzzle 1, 16, 28, 38, 49, 60, ... The maximal value Template:Math of the stepping stone puzzle Template:OEIS link

Figurate numbers

Template:Main

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

Types of primes

Template:Main

Name First elements Short description OEIS
Mersenne prime exponents 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes Template:Math such that Template:Math is prime. Template:OEIS link
Mersenne primes 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ... Template:Math is prime, where Template:Math is a prime. Template:OEIS link
Wagstaff primes 3, 11, 43, 683, 2731, 43691, ... A prime number p of the form p=2q+13 where q is an odd prime. Template:OEIS link
Wieferich primes 1093, 3511 Primes p satisfying Template:Math. Template:OEIS link
Sophie Germain primes 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ... A prime number Template:Math such that Template:Math is also prime. Template:OEIS link
Wilson primes 5, 13, 563 Primes p satisfying Template:Math. Template:OEIS link
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 Template:Math. Template:OEIS link
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). Template:OEIS link
Wolstenholme primes 16843, 2124679 Primes p satisfying (2p1p1)1(modp4). Template:OEIS link
Ramanujan primes 2, 11, 17, 29, 41, 47, 59, 67, ... The Template:Mathth Ramanujan prime is the least integer Template:Math for which Template:Math, for all Template:Math. Template:OEIS link

Base-dependent

Template:Main category

Name First elements Short description OEIS
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. Template:OEIS link
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. Template:OEIS link
Permutable primes 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ... The numbers for which every permutation of digits is a prime. Template:OEIS link
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). Template:OEIS link
Factorions 1, 2, 145, 40585, ... A natural number that equals the sum of the factorials of its decimal digits. Template:OEIS link
Circular primes 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ... The numbers which remain prime under cyclic shifts of digits. Template:OEIS link
Home prime 1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... For Template:Math is the prime that is finally reached when you start with Template:Math, concatenate its prime factors (A037276) and repeat until a prime is reached; Template:Math if no prime is ever reached. Template:OEIS link
Undulating numbers 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ... A number that has the digit form Template:Math. Template:OEIS link
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. Template:OEIS link
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). Template:OEIS link
Pandigital numbers 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, ... Numbers containing the digits Template:Math such that each digit appears exactly once. Template:OEIS link

References

Template:Series (mathematics)