List of OEIS sequences

This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their own English Wikipedia entries.

OEIS link	Name	First elements	Short description
A000002	Kolakoski sequence	{1, 2, 2, 1, 1, 2, 1, 2, 2, 1, …}	The $n$ th term describes the length of the $n$ th 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 $n$ .
A000027	Natural numbers	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}	The natural numbers (positive integers) $n \in ℕ$ .
A000032	Lucas numbers $L (n)$	{2, 1, 3, 4, 7, 11, 18, 29, 47, 76, …}	$L (n) = L (n - 1) + L (n - 2)$ for $n \geq 2$ , 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 $n \geq 1$ .
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 $n \geq 2$ , with $F (0) = 0$ and $F (1) = 1$ .
A000058	Sylvester's sequence	{2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, …}	$a (n + 1) = a (n) \cdot a (n - 1) \cdot \dots \cdot a (0) + 1 = a (n) 2 - a (n) + 1$ for $n \geq 1$ , 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 $n \geq 3$ , with $T (0) = 0 and T (1) = T (2) = 1$ .
A000108	Catalan numbers $C n$	{1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …}	$C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}},\quad n\geq 0.$
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, …}	$a (n) = 2 a (n - 1) + a (n - 2)$ for $n \geq 2$ , with $a (0) = 0, a (1) = 1$ .
A000142	Factorials $n!$	{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, …}	$n! := 1 \cdot 2 \cdot 3 \cdot 4 \cdot \dots \cdot n$ for $n \geq 1$ , with $0! = 1$ (empty product).
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, …}	$F n = 2 2 n + 1$ for $n \geq 0$ .
A000217	Triangular numbers $t (n)$	{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, …}	$t (n) = C (n + 1, 2) = n (n + 1) / 2 = 1 + 2 + \dots + n$ for $n \geq 1$ , with $t (0) = 0$ (empty sum).
A000290	Square numbers $n 2$	{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...}	$n 2 = n \times n$
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, …}	$n (n + 1)(2 n + 1) / 6$ : The number of stacked spheres in a pyramid with a square base.
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 $n$ .
A000578	Cube numbers $n 3$	{0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...}	$n 3 = n \times n \times n$
A000668	Mersenne primes	{3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, …}	$2 p - 1$ is prime, where $p$ is a prime.
A000793	Landau's function	{1, 1, 2, 3, 4, 6, 6, 12, 15, 20, …}	The largest order of permutation of $n$ elements.
A000796	Decimal expansion of $π$	{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, …}	Ratio of a circle's circumference to its diameter.
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 $n \geq 3$ , 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 $a (1) a (2) \dots a (n) + 1$ .
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.
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, …}	$a (n) = a (n - 1) + 2 a (n - 2)$ for $n \geq 2$ , 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 $n$ .
A001113	Decimal expansion of $e$	{2, 7, 1, 8, 2, 8, 1, 8, 2, 8, …}	Euler's number in base 10.
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 $2 n - 1$ nodes in all).
A001220	Wieferich primes	{1093, 3511}	Primes $p$ satisfying $2 p -1 \equiv 1 (mod p 2)$ .
A001263	Narayana numbers	{1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, …}	${1 \over k}{n-1 \choose k-1}{n \choose k-1}$ read by rows.
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 $n \geq 3$ , with $P (0) = 3, P (1) = 0, P (2) = 2$ .
A001620	Euler–Mascheroni constant $γ$	{5, 7, 7, 2, 1, 5, 6, 6, 4, 9, …}	$\gamma =\lim _{{n\rightarrow \infty }}\left(\sum _{{k=1}}^{n}{\frac {1}{k}}-\ln(n)\right)=\lim _{{b\rightarrow \infty }}\int _{1}^{b}\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.$
A001622	Decimal expansion of the golden ratio $φ$	{1, 6, 1, 8, 0, 3, 3, 9, 8, 8, …}	$φ = 1 + \sqrt 5 / 2 = 1.6180339887...$ in base 10.
A002064	Cullen numbers $C n$	{1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, …}	$C n = n \cdot 2 n + 1$ , with $n \geq 0$ .
A002110	Primorials $p n #$	{1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, …}	$p n #$ , the product of the first $n$ primes.
A002113	Palindromic numbers	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}	A number that remains the same when its digits are reversed.
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.
A002193	Decimal expansion of $\sqrt 2$	{1, 4, 1, 4, 2, 1, 3, 5, 6, 2, …}	Square root of 2.
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 $d (n) / n e \geq d (k) / k e$ for all $k > 1$ .
A002378	Pronic numbers	{0, 2, 6, 12, 20, 30, 42, 56, 72, 90, …}	$2 t (n) = n (n + 1)$ , with $n \geq 0$ .
A002559	Markov numbers	{1, 2, 5, 13, 29, 34, 89, 169, 194, …}	Positive integer solutions of $x 2 + y 2 + z 2 = 3 xyz$ .
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.
A002997	Carmichael numbers	{561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, …}	Composite numbers $n$ such that $a n - 1 \equiv 1 (mod n)$ if $a$ is prime to $n$ .
A003261	Woodall numbers	{1, 7, 23, 63, 159, 383, 895, 2047, 4607, …}	$n \cdot 2 n - 1$ , with $n \geq 1$ .
A003459	Permutable primes	{2, 3, 5, 7, 11, 13, 17, 31, 37, 71, …}	The numbers for which every permutation of digits is a prime.
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 $σ (n) < 2 n$ .
A005101	Abundant numbers	{12, 18, 20, 24, 30, 36, 40, 42, 48, 54, …}	Positive integers $n$ such that $σ (n) > 2 n$ .
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.
A005150	Look-and-say sequence	{1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, …}	A = 'frequency' followed by 'digit'-indication.
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.
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.
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).
A005384	Sophie Germain primes	{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, …}	A prime number $p$ such that $2 p + 1$ is also prime.
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.
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.
A007318	Pascal's triangle	{1, 1, 1, 1, 2, 1, 1, 3, 3, 1, …}	Pascal's triangle read by rows.
A007540	Wilson primes	{5, 13, 563}	Primes $p$ satisfying $(p -1)! \equiv -1 (mod p 2)$ .
A007588	Stella octangula numbers	{0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, …}	Stella octangula numbers: $n (2 n 2 - 1)$ , with $n \geq 0$ .
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 $1$ .
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	Prouhet–Thue–Morse constant	{0, 1, 1, 0, 1, 0, 0, 1, 1, 0, …}	$\tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}.$
A014080	Factorions	{1, 2, 145, 40585, …}	A natural number that equals the sum of the factorials of its decimal digits.
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.
A016114	Circular primes	{2, 3, 5, 7, 11, 13, 17, 37, 79, 113, …}	The numbers which remain prime under cyclic shifts of digits.
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 $σ 2 (n) = σ (σ (n)) = 2 n .$
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, …}
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.
A033307	Decimal expansion of Champernowne constant	{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, …}	Formed by concatenating the positive integers.
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.
A035513	Wythoff array	{1, 2, 4, 3, 7, 6, 5, 11, 10, 9, …}	A matrix of integers derived from the Fibonacci sequence.
A036262	Gilbreath's conjecture	{2, 1, 3, 1, 2, 5, 1, 0, 2, 7, …}	Triangle of numbers arising from Gilbreath's conjecture.
A037274	Home prime	{1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, …}	For $n \geq 2, a (n)$ 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 $ababab$ .
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.
A052486	Achilles numbers	{72, 108, 200, 288, 392, 432, 500, 648, 675, 800, …}	Positive integers which are powerful but imperfect.
A060006	Decimal expansion of Pisot–Vijayaraghavan number	{1, 3, 2, 4, 7, 1, 7, 9, 5, 7, …}	Real root of $x 3 - x - 1$ .
A076336	Sierpinski numbers	{78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, …}	Odd $k$ for which ${k \cdot 2 n + 1 : n \in ℕ}$ consists only of composite numbers.
A076337	Riesel numbers	{509203, 762701, 777149, 790841, 992077, …}	Odd $k$ for which ${k \cdot 2 n - 1 : n \in ℕ}$ 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 $a (n) = 0$ .
A088164	Wolstenholme primes	{16843, 2124679}	Primes $p$ satisfying ${2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}}$ .
A090822	Gijswijt'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 $n -1$
A094683	Juggler sequence	{0, 1, 1, 5, 2, 11, 2, 18, 2, 27, …}	If $n \equiv 0 (mod 2)$ then $⌊ \sqrt n ⌋$ 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 $k$ .
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.
A104272	Ramanujan primes	{2, 11, 17, 29, 41, 47, 59, 67, …}	The $n$ ^th Ramanujan prime is the least integer $R n$ for which $π (x) - π (x /2) \geq n$ , for all $x \geq R n$ .
A122045	Euler numbers	{1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, …}	${\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.$
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.

References

OEIS core sequences

External links

Index to OEIS

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