List of integer sequences explained

This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences.

General

Name First elements Short description OEIS
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...The th term describes the length of the th run
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ... is the number of positive integers not greater than that are coprime with .
Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... for, with and .
Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...The prime numbers, with . A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
Partition numbers
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...The partition numbers, number of additive breakdowns of n.
Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... for, with and .
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ... for, with .
Tribonacci numbers 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ... for, with .
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...Powers of 2: 2n for n ≥ 0
Polyominoes 1, 1, 1, 2, 5, 12, 35, 108, 369, ...The number of free polyominoes with cells.
Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

Cn=

1
n+1

{2n\choosen}=

(2n)!
(n+1)!n!

=

n
\prod\limits
k=2
n+k
k

,   n\ge0.

Bell numbers 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ... is the number of partitions of a set with elements.
Euler zigzag numbers 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ... is the number of linear extensions of the "zig-zag" poset.
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...The maximal number of pieces formed when slicing a pancake with cuts.
Pell numbers 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ... for, with .
Factorials 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... for, with (empty product).
Derangements 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...Number of permutations of n elements with no fixed points.
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ... is the sum of divisors of a positive integer .
Fermat numbers 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... for .
Polytrees 1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...Number of oriented trees with n nodes.
Perfect numbers 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ... is equal to the sum of the proper divisors of .
1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ...Values of the Ramanujan tau function, at n = 1, 2, 3, ...
1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...The largest order of permutation of elements.
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.
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ... for, with .
2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ... is smallest prime factor of .
Lucky numbers 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...A natural number in a set that is filtered by a sieve.
Prime powers 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...Positive integer powers of prime numbers
Central binomial coefficients 1, 2, 6, 20, 70, 252, 924, ...

{2n\choosen}=

(2n)!
(n!)2

foralln\geq0

, numbers in the center of even rows of Pascal's triangle
Motzkin numbers 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...The number of ways of drawing any number of nonintersecting chords joining (labeled) points on a circle.
Jordan–Pólya numbers 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ...Numbers that are the product of factorials.
Jacobsthal numbers 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... for, with .
0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ... is the sum of the proper divisors of the positive integer .
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 endpoints (and nodes in all).
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.
Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...Products of two primes, not necessarily distinct.
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ... is the number of times occurs, starting with .
Perrin numbers 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ... for, with .
0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, ...Used in the analysis of comparison sorts.
Cullen numbers 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ..., with .
Primorials 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ..., the product of the first primes.
Highly composite numbers 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...A positive integer with more divisors than any smaller positive integer.
Superior highly composite numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...A positive integer for which there is an such that for all .
Pronic numbers 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ..., with where are the triangular numbers.
Markov numbers 1, 2, 5, 13, 29, 34, 89, 169, 194, ...Positive integer solutions of .
Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...The numbers of the form for and .
1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... for is least number which is a unique sum of two distinct earlier terms; semiperfect.
Prime knots 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...The number of prime knots with n crossings.
Carmichael numbers 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ...Composite numbers such that if is coprime with .
Woodall numbers 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ..., with .
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.
Colossally abundant numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...A number n is colossally abundant if there is an such that for all,
\sigma(n)\geq
n1+\varepsilon
\sigma(k)
k1+\varepsilon

,

where denotes the sum-of-divisors function.
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 .
Deficient numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...Positive integers such that .
Abundant numbers 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...Positive integers such that .
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.
Recamán's sequence0, 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.
Look-and-say sequence1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...A = 'frequency' followed by 'digit'-indication.
Practical numbers1, 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.
Alternating factorial1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...n! − (n−1)+ (n−2)! − ... ± 1.
Fortunate numbers 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ...The smallest integer such that is a prime number, where the primorial is the product of the first prime numbers.
Semiperfect numbers 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ...A natural number that is equal to the sum of all or some of its proper divisors.
Magic constants 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, ...Sum of numbers in any row, column, or diagonal of a magic square of order .
Weird numbers 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...A natural number that is abundant but not semiperfect.
Farey sequence numerators0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ... 
Farey sequence denominators1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ... 
Euclid numbers 2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ..., i.e. product of first consecutive primes.
Kaprekar numbers 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ..., where and .
Sphenic numbers30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...Products of 3 distinct primes.
Giuga numbers30, 858, 1722, 66198, 2214408306, ...Composite numbers so that for each of its distinct prime factors pi we have
2
p
i

\, (n - p_i).
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...The radical of a positive integer is the product of the distinct prime numbers dividing .
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...
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.
Blum integers 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ...Numbers of the form where and are distinct primes congruent to .
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.
Superperfect numbers 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...Positive integers for which
Bernoulli numbers 1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ... 
Hyperperfect numbers6, 21, 28, 301, 325, 496, 697, ...-hyperperfect numbers, i.e. for which the equality holds.
Achilles numbers 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...Positive integers which are powerful but imperfect.
Primary pseudoperfect numbers 2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...Satisfies a certain Egyptian fraction.
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.
Sierpinski numbers 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...Odd for which consists only of composite numbers.
Riesel numbers 509203, 762701, 777149, 790841, 992077, ...Odd for which consists only of composite numbers.
1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ... if the binary representation of contains no block of consecutive zeros of odd length; otherwise .
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ...The th term counts the maximal number of repeated blocks at the end of the subsequence from to
Carol numbers −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...

a(n)=(2n-1)2-2.

0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...If then else .
Highly totient numbers 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...Each number on this list has more solutions to the equation than any preceding .
Euler numbers 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...
1
\cosht

=

2
et+e-t

=

infty
\sum
n=0
En
n!

⋅ tn.

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.
Erdős–Nicolas numbers 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...A number such that there exists another number and

\sumdd=n.

Solution to Stepping Stone Puzzle 1, 16, 28, 38, 49, 60, ...The maximal value of the stepping stone puzzle

Figurate numbers

See main article: Figurate number.

Name First elements Short description OEIS
Natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...The natural numbers (positive integers) .
Triangular numbers 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... for, with (empty sum).
Square numbers 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...
Tetrahedral numbers 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ... is the sum of the first triangular numbers, with (empty sum).
Square pyramidal numbers 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ...

The number of stacked spheres in a pyramid with a square base.

Cube numbers 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...
Fifth powers 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...
Star numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...Sn = 6n(n − 1) + 1.
Stella octangula numbers 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, ...Stella octangula numbers:, with .

Types of primes

See main article: List of prime numbers.

Name First elements Short description OEIS
Mersenne prime exponents 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...Primes such that is prime.
Mersenne primes3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ... is prime, where is a prime.
Wagstaff primes3, 11, 43, 683, 2731, 43691, ...A prime number p of the form

p={{2q+1}\over3}

where q is an odd prime.
Wieferich primes 1093, 3511Primes

p

satisfying .
Sophie Germain primes 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ...A prime number such that is also prime.
Wilson primes 5, 13, 563Primes

p

satisfying .
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 .
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).
Wolstenholme primes 16843, 2124679Primes

p

satisfying

{2p-1\choosep-1}\equiv1\pmod{p4}

.
Ramanujan primes 2, 11, 17, 29, 41, 47, 59, 67, ...The th Ramanujan prime is the least integer for which, for all .

Base-dependent

See main article: category.

Name First elements Short description OEIS
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.
Palindromic numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ...A number that remains the same when its digits are reversed.
Permutable primes 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ...The numbers for which every permutation of digits is a prime.
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).
Factorions 1, 2, 145, 40585, ...A natural number that equals the sum of the factorials of its decimal digits.
Circular primes 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ...The numbers which remain prime under cyclic shifts of digits.
1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ...For is the prime that is finally reached when you start with, concatenate its prime factors (A037276) and repeat until a prime is reached; if no prime is ever reached.
Undulating numbers 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ...A number that has the digit form .
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.
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).
Pandigital numbers 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, ...Numbers containing the digits such that each digit appears exactly once.

References

External links

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