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 $n$ .
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).
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 $n$ .
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).
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.
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 $a (n) = 0$ .
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, …}	${\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.$

References

OEIS core sequences

External links

Index to OEIS

Sequences and series

Integer
sequences

Basic	Arithmetic progression Geometric progression Square number Cubic number Factorial Powers of two Powers of 10

Advanced	Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number others

Properties of sequences

Properties of series

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/1^s + 1/2^s+ 1/3^s + ... (Riemann zeta function)

Divergent	1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric
series

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