This is a list of notable integer sequences.
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 p_{n} | {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...} | The prime numbers p_{n}, 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 P_{n} |
{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)^{2} − a(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: 2^{n} 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 C_{n} | {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...} | |
A000110 | Bell numbers B_{n} | {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...} | B_{n} is the number of partitions of a set with n elements. |
A000111 | Euler zigzag numbers E_{n} | {1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ...} | E_{n} 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 P_{n} | {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) := σ_{1}(n) is the sum of divisors of a positive integer n. |
A000215 | Fermat numbers F_{n} | {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...} | F_{n} = 2^{2n} + 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, ...} | , 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 P_{n} | {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 C_{n} | {1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...} | C_{n} = n⋅2^{n} + 1, with n ≥ 0. |
A002110 | Primorials p_{n}# | {1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...} | p_{n}#, 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)/n^{e} ≥ d(k)/k^{e} 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 x^{2} + y^{2} + z^{2} = 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 a^{n − 1} ≡ 1 (mod n) if a is coprime with n. |
A003261 | Woodall numbers | {1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...} | n⋅2^{n} − 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,
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 p_{n}# + m is a prime number, where the primorial p_{n}# 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, ...} | p_{n}# + 1, i.e. 1 + product of first n consecutive primes. |
A006886 | Kaprekar numbers | {1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...} | X^{2} = Ab^{n} + B, where 0 < B < b^{n} 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 p_{i} we have . |
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 σ^{2}(n) = σ(σ(n)) = 2n. |
A027641 | Bernoulli numbers B_{n} | {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⋅2^{n} + 1 : n ∈ } consists only of composite numbers. |
A076337 | Riesel numbers | {509203, 762701, 777149, 790841, 992077, ...} | Odd k for which { k⋅2^{n} − 1 : 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, ...} | |
A094683 | Juggler sequence | {0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...} | If n ≡ 0 (mod 2) then ⌊√n⌋ else ⌊n^{3/2}⌋. |
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, ...} | |
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 |
A337663 | Solution to Stepping Stone Puzzle | {1, 16, 28, 38, 49, 60 ...} | The maximal value a(n) of the stepping stone puzzle |
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 ∈ . |
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 n^{2} | {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...} | n^{2} = 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 n^{3} | {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...} | n^{3} = n × n × n |
A000584 | Fifth powers | {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...} | n^{5} |
A003154 | Star numbers | {1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...} | S_{n} = 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(2n^{2} − 1), with n ≥ 0. |
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 2^{p} − 1 is prime. |
A000668 | Mersenne primes | {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ...} | 2^{p} − 1 is prime, where p is a prime. |
A000979 | Wagstaff primes | {3, 11, 43, 683, 2731, 43691, ...} | A prime number p of the form where q is an odd prime. |
A001220 | Wieferich primes | {1093, 3511} | Primes satisfying 2^{p−1} ≡ 1 (mod p^{2}). |
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 satisfying (p−1)! ≡ −1 (mod p^{2}). |
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 satisfying . |
A104272 | Ramanujan primes | {2, 11, 17, 29, 41, 47, 59, 67, ...} | The n^{th} Ramanujan prime is the least integer R_{n} for which π(x) − π(x/2) ≥ n, for all x ≥ R_{n}. |
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. |