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This is a list of integer sequences in the On-Line Encyclopedia of 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 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)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. |
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! := 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) := 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 = n⋅2n + 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)/ne≥d(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, ...} | n⋅2n− 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 { k⋅2n + 1 : n∈ ℕ } consists only of composite numbers. |
A076337 | Riesel numbers | {509203, 762701, 777149, 790841, 992077, ...} | Odd k for which { k⋅2n− 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 |
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 ∈ 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. |
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 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 x ≥ Rn. |
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, ...} | 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. |
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.