List of OEIS sequences

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This is a list of integer sequences in the On-Line Encyclopedia of Integer Sequences.

Contents

General

OEIS linkNameFirst elementsShort 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 prime to 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.
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.
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, ...}
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! := 1234n 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.
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.
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 = n2n + 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, ...}2t(n) = n (n + 1), with n 0.
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 prime to n.
A003261 Woodall numbers {1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...}n2n 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 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, ...}Sum of numbers in any row, column, or diagonal of a magic square of order n = 3, 4, 5, 6, 7, 8, ....
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.
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.
A054037 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 { k2n + 1 : n ℕ } consists only of composite numbers.
A076337 Riesel numbers {509203, 762701, 777149, 790841, 992077, ...}Odd k for which { k2n 1 : n 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 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, ...}
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

Figurate numbers

OEIS linkNameFirst elementsShort description
A000027 Natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}The natural numbers (positive integers) n ∈ 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, ...}The nth star number is 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 linkNameFirst elementsShort 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 where q is an odd prime.
A001220 Wieferich primes {1093, 3511}Primes 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 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 satisfying .
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 linkNameFirst elementsShort 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, ...}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 09 such that each digit appears exactly once.

Related Research Articles

79 (seventy-nine) is the natural number following 78 and preceding 80.

68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number.

109 is the natural number following 108 and preceding 110.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it is often written with a comma separating the thousands unit: 1,000.

300 is the natural number following 299 and preceding 301.

600 is the natural number following 599 and preceding 601.

800 is the natural number following 799 and preceding 801.

900 is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 integers. In base 10 it is a Harshad number.

2000 is a natural number following 1999 and preceding 2001.

126 is the natural number following 125 and preceding 127.

229 is the natural number following 228 and preceding 230.

227 is the natural number between 226 and 228. It is also a prime number.

225 is the natural number following 224 and preceding 226.

233 is the natural number following 232 and preceding 234.

263 is the natural number between 262 and 264. It is also a prime number.

277 is the natural number following 276 and preceding 278.

204 is the natural number following 203 and preceding 205.

232 is the natural number following 231 and preceding 233.

252 is the natural number following 251 and preceding 253.

271 is the natural number after 270 and before 272.

References