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List of integer sequences

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

General edit

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 A000002
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. A000010
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. A000032
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. A000040
Partition numbers
Pn
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... The partition numbers, number of additive breakdowns of n. A000041
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. A000045
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. A000058
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. A000073
Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... Powers of 2: 2n for n ≥ 0 A000079
Polyominoes 1, 1, 1, 2, 5, 12, 35, 108, 369, ... The number of free polyominoes with n cells. A000105
Catalan numbers Cn 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...   A000108
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. A000110
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. A000111
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. A000124
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. A000129
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). A000142
Derangements 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ... Number of permutations of n elements with no fixed points. A000166
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. A000203
Fermat numbers Fn 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... Fn = 22n + 1 for n ≥ 0. A000215
Polytrees 1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ... Number of oriented trees with n nodes. A000238
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. A000396
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, ... A000594
Landau's function 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ... The largest order of permutation of n elements. A000793
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. A000930
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. A000931
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. A000945
Lucky numbers 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ... A natural number in a set that is filtered by a sieve. A000959
Prime powers 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ... Positive integer powers of prime numbers A000961
Central binomial coefficients 1, 2, 6, 20, 70, 252, 924, ...  , numbers in the center of even rows of Pascal's triangle A000984
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. A001006
Jordan–Pólya numbers 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ... Numbers that are the product of factorials. A001013
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. A001045
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. A001065
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). A001190
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. A001316
Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... Products of two primes, not necessarily distinct. A001358
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. A001462
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. A001608
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. A001855
Cullen numbers Cn 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ... Cn = n⋅2n + 1, with n ≥ 0. A002064
Primorials pn# 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ... pn#, the product of the first n primes. A002110
Highly composite numbers 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... A positive integer with more divisors than any smaller positive integer. A002182
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. A002201
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. A002378
Markov numbers 1, 2, 5, 13, 29, 34, 89, 169, 194, ... Positive integer solutions of x2 + y2 + z2 = 3xyz. A002559
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. A002808
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. A002858
Prime knots 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ... The number of prime knots with n crossings. A002863
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. A002997
Woodall numbers 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ... n⋅2n − 1, with n ≥ 1. A003261
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. A003601
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,
 

where σ denotes the sum-of-divisors function.

A004490
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. A005044
Deficient numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ... Positive integers n such that σ(n) < 2n. A005100
Abundant numbers 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ... Positive integers n such that σ(n) > 2n. A005101
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. A005114
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. A005132
Look-and-say sequence 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ... A = 'frequency' followed by 'digit'-indication. A005150
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. A005153
Alternating factorial 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ... n! − (n−1)! + (n−2)! − ... ± 1!. A005165
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. A005235
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. A005835
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. A006003
Weird numbers 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ... A natural number that is abundant but not semiperfect. A006037
Farey sequence numerators 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...   A006842
Farey sequence denominators 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...   A006843
Euclid numbers 2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ... pn# + 1, i.e. 1 + product of first n consecutive primes. A006862
Kaprekar numbers 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ... X2 = Abn + B, where 0 < B < bn and X = A + B. A006886
Sphenic numbers 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ... Products of 3 distinct primes. A007304
Giuga numbers 30, 858, 1722, 66198, 2214408306, ... Composite numbers so that for each of its distinct prime factors pi we have  . A007850
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. A007947
Thue–Morse sequence 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ... A010060
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. A014577
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). A016105
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. A018226
Superperfect numbers 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ... Positive integers n for which σ2(n) = σ(σ(n)) = 2n. A019279
Bernoulli numbers Bn 1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...   A027641
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. A034897
Achilles numbers 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ... Positive integers which are powerful but imperfect. A052486
Primary pseudoperfect numbers 2, 6, 42, 1806, 47058, 2214502422, 52495396602, ... Satisfies a certain Egyptian fraction. A054377
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. A059756
Sierpinski numbers 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ... Odd k for which { k⋅2n + 1 : n  } consists only of composite numbers. A076336
Riesel numbers 509203, 762701, 777149, 790841, 992077, ... Odd k for which { k⋅2n − 1 : n  } consists only of composite numbers. A076337
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. A086747
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 A090822
Carol numbers −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...   A093112
Juggler sequence 0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ... If n ≡ 0 (mod 2) then n else n3/2. A094683
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. A097942
Euler numbers 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...   A122045
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. A138591
Erdős–Nicolas numbers 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ... A number n such that there exists another number m and   A194472
Solution to Stepping Stone Puzzle 1, 16, 28, 38, 49, 60, ... The maximal value a(n) of the stepping stone puzzle A337663

Figurate numbers edit

Name First elements Short description OEIS
Natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... The natural numbers (positive integers) n . A000027
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). A000217
Square numbers n2 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ... n2 = n × n A000290
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). A000292
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. A000330
Cube numbers n3 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ... n3 = n × n × n A000578
Fifth powers 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ... n5 A000584
Star numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ... Sn = 6n(n − 1) + 1. A003154
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. A007588

Types of primes edit

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. A000043
Mersenne primes 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ... 2p − 1 is prime, where p is a prime. A000668
Wagstaff primes 3, 11, 43, 683, 2731, 43691, ... A prime number p of the form   where q is an odd prime. A000979
Wieferich primes 1093, 3511 Primes   satisfying 2p−1 ≡ 1 (mod p2). A001220
Sophie Germain primes 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ... A prime number p such that 2p + 1 is also prime. A005384
Wilson primes 5, 13, 563 Primes   satisfying (p−1)! ≡ −1 (mod p2). A007540
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. A007770
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). A088054
Wolstenholme primes 16843, 2124679 Primes   satisfying  . A088164
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. A104272

Base-dependent edit

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. A005224
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. A002113
Permutable primes 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ... The numbers for which every permutation of digits is a prime. A003459
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). A005349
Factorions 1, 2, 145, 40585, ... A natural number that equals the sum of the factorials of its decimal digits. A014080
Circular primes 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ... The numbers which remain prime under cyclic shifts of digits. A016114
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. A037274
Undulating numbers 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ... A number that has the digit form ababab. A046075
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. A046758
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). A046760
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. A050278

References edit

  • OEIS core sequences

External links edit

  • Index to OEIS

list, integer, sequences, this, list, notable, integer, sequences, with, links, their, entries, line, encyclopedia, integer, sequences, contents, general, figurate, numbers, types, primes, base, dependent, references, external, linksgeneral, editname, first, e. This is a list of notable integer sequences with links to their entries in the On Line Encyclopedia of Integer Sequences Contents 1 General 2 Figurate numbers 3 Types of primes 4 Base dependent 5 References 6 External linksGeneral editName First elements Short description OEISKolakoski sequence 1 2 2 1 1 2 1 2 2 1 The n th term describes the length of the n th run A000002Euler s totient function f n 1 1 2 2 4 2 6 4 6 4 f n is the number of positive integers not greater than n that are coprime with n A000010Lucas 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 A000032Prime 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 A000040Partition numbers Pn 1 1 2 3 5 7 11 15 22 30 42 The partition numbers number of additive breakdowns of n A000041Fibonacci 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 A000045Sylvester s sequence 2 3 7 43 1807 3263443 10650056950807 113423713055421844361000443 a n 1 a n a n 1 a 0 1 a n 2 a n 1 for n 1 with a 0 2 A000058Tribonacci 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 A000073Powers of 2 1 2 4 8 16 32 64 128 256 512 1024 Powers of 2 2n for n 0 A000079Polyominoes 1 1 1 2 5 12 35 108 369 The number of free polyominoes with n cells A000105Catalan numbers Cn 1 1 2 5 14 42 132 429 1430 4862 Cn 1n 1 2nn 2n n 1 n k 2nn kk n 0 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 nbsp A000108Bell numbers Bn 1 1 2 5 15 52 203 877 4140 21147 Bn is the number of partitions of a set with n elements A000110Euler 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 A000111Lazy 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 A000124Pell 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 A000129Factorials 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 A000142Derangements 1 0 1 2 9 44 265 1854 14833 133496 1334961 14684570 176214841 Number of permutations of n elements with no fixed points A000166Divisor function s n 1 3 4 7 6 12 8 15 13 18 12 28 s n s1 n is the sum of divisors of a positive integer n A000203Fermat numbers Fn 3 5 17 257 65537 4294967297 18446744073709551617 340282366920938463463374607431768211457 Fn 22n 1 for n 0 A000215Polytrees 1 1 3 8 27 91 350 1376 5743 24635 108968 Number of oriented trees with n nodes A000238Perfect numbers 6 28 496 8128 33550336 8589869056 137438691328 2305843008139952128 n is equal to the sum s n s n n of the proper divisors of n A000396Ramanujan tau function 1 24 252 1472 4830 6048 16744 84480 113643 Values of the Ramanujan tau function t n at n 1 2 3 A000594Landau s function 1 1 2 3 4 6 6 12 15 20 The largest order of permutation of n elements A000793Narayana 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 A000930Padovan 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 A000931Euclid 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 A000945Lucky numbers 1 3 7 9 13 15 21 25 31 33 A natural number in a set that is filtered by a sieve A000959Prime powers 2 3 4 5 7 8 9 11 13 16 17 19 Positive integer powers of prime numbers A000961Central binomial coefficients 1 2 6 20 70 252 924 2nn 2n n 2 for all n 0 displaystyle 2n choose n frac 2n n 2 text for all n geq 0 nbsp numbers in the center of even rows of Pascal s triangle A000984Motzkin 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 A001006Jordan Polya numbers 1 2 4 6 8 12 16 24 32 36 48 64 Numbers that are the product of factorials A001013Jacobsthal 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 A001045Sum of proper divisors s n 0 1 1 3 1 6 1 7 4 8 s n s n n is the sum of the proper divisors of the positive integer n A001065Wedderburn 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 A001190Gould 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 A001316Semiprimes 4 6 9 10 14 15 21 22 25 26 Products of two primes not necessarily distinct A001358Golomb 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 A001462Perrin 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 A001608Sorting number 0 1 3 5 8 11 14 17 21 25 29 33 37 41 45 49 Used in the analysis of comparison sorts A001855Cullen numbers Cn 1 3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 Cn n 2n 1 with n 0 A002064Primorials pn 1 2 6 30 210 2310 30030 510510 9699690 223092870 pn the product of the first n primes A002110Highly composite numbers 1 2 4 6 12 24 36 48 60 120 A positive integer with more divisors than any smaller positive integer A002182Superior highly composite numbers 2 6 12 60 120 360 2520 5040 55440 720720 A positive integer n for which there is an e gt 0 such that d n ne d k ke for all k gt 1 A002201Pronic 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 A002378Markov numbers 1 2 5 13 29 34 89 169 194 Positive integer solutions of x2 y2 z2 3xyz A002559Composite numbers 4 6 8 9 10 12 14 15 16 18 The numbers n of the form xy for x gt 1 and y gt 1 A002808Ulam number 1 2 3 4 6 8 11 13 16 18 a 1 1 a 2 2 for n gt 2 a n is least number gt a n 1 which is a unique sum of two distinct earlier terms semiperfect A002858Prime knots 0 0 1 1 2 3 7 21 49 165 552 2176 9988 The number of prime knots with n crossings A002863Carmichael 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 A002997Woodall numbers 1 7 23 63 159 383 895 2047 4607 n 2n 1 with n 1 A003261Arithmetic 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 A003601Colossally abundant numbers 2 6 12 60 120 360 2520 5040 55440 720720 A number n is colossally abundant if there is an e gt 0 such that for all k gt 1 s n n1 e s k k1 e displaystyle frac sigma n n 1 varepsilon geq frac sigma k k 1 varepsilon nbsp where s denotes the sum of divisors function A004490Alcuin 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 A005044Deficient numbers 1 2 3 4 5 7 8 9 10 11 Positive integers n such that s n lt 2n A005100Abundant numbers 12 18 20 24 30 36 40 42 48 54 Positive integers n such that s n gt 2n A005101Untouchable 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 A005114Recaman 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 gt 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 A005132Look and say sequence 1 11 21 1211 111221 312211 13112221 1113213211 31131211131221 13211311123113112211 A frequency followed by digit indication A005150Practical 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 A005153Alternating factorial 1 1 5 19 101 619 4421 35899 326981 3301819 36614981 442386619 5784634181 81393657019 n n 1 n 2 1 A005165Fortunate numbers 3 5 7 13 23 17 19 23 37 61 The smallest integer m gt 1 such that pn m is a prime number where the primorial pn is the product of the first n prime numbers A005235Semiperfect 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 A005835Magic 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 A006003Weird numbers 70 836 4030 5830 7192 7912 9272 10430 10570 10792 A natural number that is abundant but not semiperfect A006037Farey sequence numerators 0 1 0 1 1 0 1 1 2 1 A006842Farey sequence denominators 1 1 1 2 1 1 3 2 3 1 A006843Euclid numbers 2 3 7 31 211 2311 30031 510511 9699691 223092871 pn 1 i e 1 product of first n consecutive primes A006862Kaprekar numbers 1 9 45 55 99 297 703 999 2223 2728 X2 Abn B where 0 lt B lt bn and X A B A006886Sphenic numbers 30 42 66 70 78 102 105 110 114 130 Products of 3 distinct primes A007304Giuga numbers 30 858 1722 66198 2214408306 Composite numbers so that for each of its distinct prime factors pi we have pi2 n pi displaystyle p i 2 n p i nbsp A007850Radical 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 A007947Thue Morse sequence 0 1 1 0 1 0 0 1 1 0 A010060Regular 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 A014577Blum 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 A016105Magic 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 A018226Superperfect numbers 2 4 16 64 4096 65536 262144 1073741824 1152921504606846976 309485009821345068724781056 Positive integers n for which s2 n s s n 2n A019279Bernoulli numbers Bn 1 1 1 0 1 0 1 0 1 0 5 0 691 0 7 0 3617 0 43867 0 A027641Hyperperfect numbers 6 21 28 301 325 496 697 k hyperperfect numbers i e n for which the equality n 1 k s n n 1 holds A034897Achilles numbers 72 108 200 288 392 432 500 648 675 800 Positive integers which are powerful but imperfect A052486Primary pseudoperfect numbers 2 6 42 1806 47058 2214502422 52495396602 Satisfies a certain Egyptian fraction A054377Erdos 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 A059756Sierpinski numbers 78557 271129 271577 322523 327739 482719 575041 603713 903983 934909 Odd k for which k 2n 1 n N displaystyle mathbb N nbsp consists only of composite numbers A076336Riesel numbers 509203 762701 777149 790841 992077 Odd k for which k 2n 1 n N displaystyle mathbb N nbsp consists only of composite numbers A076337Baum 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 A086747Gijswijt s sequence 1 1 2 1 1 2 2 2 3 1 The n th term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n 1 A090822Carol numbers 1 7 47 223 959 3967 16127 65023 261119 1046527 a n 2n 1 2 2 displaystyle a n 2 n 1 2 2 nbsp A093112Juggler sequence 0 1 1 5 2 11 2 18 2 27 If n 0 mod 2 then n else n3 2 A094683Highly totient numbers 1 2 4 8 12 24 48 72 144 240 Each number k on this list has more solutions to the equation f x k than any preceding k A097942Euler numbers 1 0 1 0 5 0 61 0 1385 0 1cosh t 2et e t n 0 Enn tn displaystyle frac 1 cosh t frac 2 e t e t sum n 0 infty frac E n n cdot t n nbsp A122045Polite 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 A138591Erdos Nicolas numbers 24 2016 8190 42336 45864 392448 714240 1571328 A number n such that there exists another number m and d n d md n displaystyle sum d mid n d leq m d n nbsp A194472Solution to Stepping Stone Puzzle 1 16 28 38 49 60 The maximal value a n of the stepping stone puzzle A337663Figurate numbers editMain article Figurate number Name First elements Short description OEISNatural numbers 1 2 3 4 5 6 7 8 9 10 The natural numbers positive integers n N displaystyle mathbb N nbsp A000027Triangular 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 A000217Square numbers n2 0 1 4 9 16 25 36 49 64 81 n2 n n A000290Tetrahedral 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 A000292Square 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 A000330Cube numbers n3 0 1 8 27 64 125 216 343 512 729 n3 n n n A000578Fifth powers 0 1 32 243 1024 3125 7776 16807 32768 59049 100000 n5 A000584Star numbers 1 13 37 73 121 181 253 337 433 541 661 793 937 Sn 6n n 1 1 A003154Stella 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 A007588Types of primes editMain article List of prime numbers Name First elements Short description OEISMersenne prime exponents 2 3 5 7 13 17 19 31 61 89 Primes p such that 2p 1 is prime A000043Mersenne primes 3 7 31 127 8191 131071 524287 2147483647 2305843009213693951 618970019642690137449562111 2p 1 is prime where p is a prime A000668Wagstaff primes 3 11 43 683 2731 43691 A prime number p of the form p 2q 13 displaystyle p 2 q 1 over 3 nbsp where q is an odd prime A000979Wieferich primes 1093 3511 Primes p displaystyle p nbsp satisfying 2p 1 1 mod p2 A001220Sophie Germain primes 2 3 5 11 23 29 41 53 83 89 A prime number p such that 2p 1 is also prime A005384Wilson primes 5 13 563 Primes p displaystyle p nbsp satisfying p 1 1 mod p2 A007540Happy 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 A007770Factorial primes 2 3 5 7 23 719 5039 39916801 A prime number that is one less or one more than a factorial all factorials gt 1 are even A088054Wolstenholme primes 16843 2124679 Primes p displaystyle p nbsp satisfying 2p 1p 1 1 modp4 displaystyle 2p 1 choose p 1 equiv 1 pmod p 4 nbsp A088164Ramanujan primes 2 11 17 29 41 47 59 67 The n th Ramanujan prime is the least integer Rn for which p x p x 2 n for all x Rn A104272Base dependent editMain category Base dependent integer sequences Name First elements Short description OEISAronson 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 A005224Palindromic 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 A002113Permutable primes 2 3 5 7 11 13 17 31 37 71 The numbers for which every permutation of digits is a prime A003459Harshad 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 A005349Factorions 1 2 145 40585 A natural number that equals the sum of the factorials of its decimal digits A014080Circular primes 2 3 5 7 11 13 17 37 79 113 The numbers which remain prime under cyclic shifts of digits A016114Home 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 A037274Undulating numbers 101 121 131 141 151 161 171 181 191 202 A number that has the digit form ababab A046075Equidigital 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 A046758Extravagant 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 A046760Pandigital numbers 1023456789 1023456798 1023456879 1023456897 1023456978 1023456987 1023457689 1023457698 1023457869 1023457896 Numbers containing the digits 0 9 such that each digit appears exactly once A050278References editOEIS core sequencesExternal links editIndex to OEIS Retrieved from https en wikipedia org w index php title List of integer sequences amp oldid 1214580353, wikipedia, wiki, book, books, library,

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