List of OEIS sequences

Table columns:
OEIS link
Name
First elements
Short description

Table rows:
cells: A000002
Kolakoski sequence
{1, 2, 2, 1, 1, 2, 1, 2, 2, 1, …}
The nth term describes the length of the nth run
cells: 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.
cells: A000027
Natural numbers
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}
The natural numbers (positive integers) n ∈ ℕ.
cells: 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.
cells: A000040
Prime numbers pn
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}
The prime numbers pn, with n ≥ 1.
cells: A000041
Partition numbers Pn
{1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...}
The partition numbers, number of additive breakdowns of n.
cells: 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.
cells: 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.
cells: 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.
cells: A000108
Catalan numbers Cn
{1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …}
C  n  =  1  n  +  1  (  2  n  n  )  =  (  2  n  )  !  (  n  +  1  )  !  n  !  =  ∏  k  =  2  n  n  +  k  k  ,  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.}
cells: 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.
cells: 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.
cells: 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.
cells: 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.
cells: 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).
cells: 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.
cells: A000215
Fermat numbers Fn
{3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, …}
Fn = 22n + 1 for n ≥ 0.
cells: 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).
cells: A000290
Square numbers n2
{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...}
n2 = n × n
cells: 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).
cells: 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.
cells: 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.
cells: A000578
Cube numbers n3
{0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...}
n3 = n × n × n
cells: A000668
Mersenne primes
{3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, …}
2p − 1 is prime, where p is a prime.
cells: A000793
Landau's function
{1, 1, 2, 3, 4, 6, 6, 12, 15, 20, …}
The largest order of permutation of n elements.
cells: A000796
Decimal expansion of π
{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, …}
Ratio of a circle's circumference to its diameter.
cells: 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.
cells: 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.
cells: 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.
cells: 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.
cells: 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.
cells: 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.
cells: 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.
cells: A001113
Decimal expansion of e
{2, 7, 1, 8, 2, 8, 1, 8, 2, 8, …}
Euler's number in base 10.
cells: 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).
cells: A001263
Narayana numbers
{1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, …}
1  k  (  n  −  1  k  −  1  )  (  n  k  −  1  )  {\displaystyle {1 \over k}{n-1 \choose k-1}{n \choose k-1}}  read by rows.
cells: A001358
Semiprimes
{4, 6, 9, 10, 14, 15, 21, 22, 25, 26, …}
Products of two primes, not necessarily distinct.
cells: 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.
cells: 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.
cells: A001620
Euler–Mascheroni constant γ
{5, 7, 7, 2, 1, 5, 6, 6, 4, 9, …}
γ  =  lim  n  →  ∞  (  ∑  k  =  1  n  1  k  −  ln  ⁡  (  n  )  )  =  lim  b  →  ∞  ∫  1  b  (  1  ⌊  x  ⌋  −  1  x  )  d  x  .  {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\lim _{b\rightarrow \infty }\int _{1}^{b}\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.}
cells: A001622
Decimal expansion of the golden ratio φ
{1, 6, 1, 8, 0, 3, 3, 9, 8, 8, …}
φ = 1 + √5/2 = 1.6180339887... in base 10.
cells: 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.
cells: A002110
Primorials pn#
{1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, …}
pn#, the product of the first n primes.
cells: A002113
Palindromic numbers
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}
A number that remains the same when its digits are reversed.
cells: 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.
cells: A002193
Decimal expansion of √2
{1, 4, 1, 4, 2, 1, 3, 5, 6, 2, …}
Square root of 2.
cells: 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.
cells: A002378
Pronic numbers
{0, 2, 6, 12, 20, 30, 42, 56, 72, 90, …}
2t(n) = n (n + 1), with n ≥ 0.
cells: A002559
Markov numbers
{1, 2, 5, 13, 29, 34, 89, 169, 194, …}
Positive integer solutions of x2 + y2 + z2 = 3xyz.
cells: 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.
cells: 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.
cells: 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.
cells: A003261
Woodall numbers
{1, 7, 23, 63, 159, 383, 895, 2047, 4607, …}
n⋅2n − 1, with n ≥ 1.
cells: A003459
Permutable primes
{2, 3, 5, 7, 11, 13, 17, 31, 37, 71, …}
The numbers for which every permutation of digits is a prime.
cells: 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.
cells: A005100
Deficient numbers
{1, 2, 3, 4, 5, 7, 8, 9, 10, 11, …}
Positive integers n such that σ(n) < 2n.
cells: A005101
Abundant numbers
{12, 18, 20, 24, 30, 36, 40, 42, 48, 54, …}
Positive integers n such that σ(n) > 2n.
cells: 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.
cells: A005150
Look-and-say sequence
{1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, …}
A = 'frequency' followed by 'digit'-indication.
cells: 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.
cells: 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.
cells: 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).
cells: 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.
cells: 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.
cells: A006037
Weird numbers
{70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, …}
A natural number that is abundant but not semiperfect.
cells: A006842
Farey sequence numerators
{0, 1, 0, 1, 1, 0, 1, 1, 2, 1, …}
cells: A006843
Farey sequence denominators
{1, 1, 1, 2, 1, 1, 3, 2, 3, 1, …}
cells: A006862
Euclid numbers
{2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, …}
pn# + 1, i.e. 1 + product of first n consecutive primes.
cells: A006886
Kaprekar numbers
{1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, …}
X2 = Abn + B, where 0 < B < bn and X = A + B.
cells: A007304
Sphenic numbers
{30, 42, 66, 70, 78, 102, 105, 110, 114, 130, …}
Products of 3 distinct primes.
cells: A007318
Pascal's triangle
{1, 1, 1, 1, 2, 1, 1, 3, 3, 1, …}
Pascal's triangle read by rows.
cells: 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.
cells: 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.
cells: 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.
cells: A010060
Prouhet–Thue–Morse constant
{0, 1, 1, 0, 1, 0, 0, 1, 1, 0, …}
τ  =  ∑  i  =  0  ∞  t  i  2  i  +  1  .  {\displaystyle \tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}.}
cells: A014080
Factorions
{1, 2, 145, 40585, …}
A natural number that equals the sum of the factorials of its decimal digits.
cells: 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.
cells: A016114
Circular primes
{2, 3, 5, 7, 11, 13, 17, 37, 79, 113, …}
The numbers which remain prime under cyclic shifts of digits.
cells: 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.
cells: A019279
Superperfect numbers
{2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, …}
Positive integers n for which σ2(n) = σ(σ(n)) = 2n.
cells: A027641
Bernoulli numbers Bn
{1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, …}
cells: A031214
First elements in all OEIS sequences
{1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …}
One of sequences referring to the OEIS itself.
cells: A033307
Decimal expansion of Champernowne constant
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, …}
Formed by concatenating the positive integers.
cells: 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.
cells: A035513
Wythoff array
{1, 2, 4, 3, 7, 6, 5, 11, 10, 9, …}
A matrix of integers derived from the Fibonacci sequence.
cells: A036262
Gilbreath's conjecture
{2, 1, 3, 1, 2, 5, 1, 0, 2, 7, …}
Triangle of numbers arising from Gilbreath's conjecture.
cells: 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.
cells: A046075
Undulating numbers
{101, 121, 131, 141, 151, 161, 171, 181, 191, 202, …}
A number that has the digit form ababab.
cells: 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.
cells: A052486
Achilles numbers
{72, 108, 200, 288, 392, 432, 500, 648, 675, 800, …}
Positive integers which are powerful but imperfect.
cells: A060006
Decimal expansion of Pisot–Vijayaraghavan number
{1, 3, 2, 4, 7, 1, 7, 9, 5, 7, …}
Real root of x3 − x − 1.
cells: 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.
cells: A076337
Riesel numbers
{509203, 762701, 777149, 790841, 992077, …}
Odd k for which { k⋅2n − 1 : n ∈ ℕ } consists only of composite numbers.
cells: 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.
cells: 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
cells: A094683
Juggler sequence
{0, 1, 1, 5, 2, 11, 2, 18, 2, 27, …}
If n ≡ 0 (mod 2) then ⌊√n⌋ else ⌊n3/2⌋.
cells: 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.
cells: A100264
Decimal expansion of Chaitin's constant
{0, 0, 7, 8, 7, 4, 9, 9, 6, 9, …}
Chaitin constant (Chaitin omega number) or halting probability.
cells: 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.
cells: A122045
Euler numbers
{1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, …}
1  cosh  ⁡  t  =  2  e  t  +  e  −  t  =  ∑  n  =  0  ∞  E  n  n  !  ⋅  t  n  .  {\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.}
cells: A1238591
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.

Questions about this table:
originalText: what is the sequence of 1 2 6 24
answer: answerTexts: Factorials n!