| Name |
First elements |
Short description |
OEIS
|
| Kolakoski sequence
|
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...
|
The Template:Mathth term describes the length of the Template:Mathth run
|
Template:OEIS link
|
| Euler's totient function Template:Math
|
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...
|
Template:Math is the number of positive integers not greater than Template:Math that are coprime with Template:Math.
|
Template:OEIS link
|
| Lucas numbers Template:Math
|
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...
|
Template:Math for Template:Math, with Template:Math and Template:Math.
|
Template:OEIS link
|
| Prime numbers Template:Math
|
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
|
The prime numbers Template:Math, with Template:Math. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.
|
Template:OEIS link
|
Partition numbers Template:Math
|
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...
|
The partition numbers, number of additive breakdowns of n.
|
Template:OEIS link
|
| Fibonacci numbers Template:Math
|
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
|
Template:Math for Template:Math, with Template:Math and Template:Math.
|
Template:OEIS link
|
| Sylvester's sequence
|
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...
|
for Template:Math, with Template:Math.
|
Template:OEIS link
|
| Tribonacci numbers
|
0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...
|
Template:Math for Template:Math, with Template:Math.
|
Template:OEIS link
|
| Powers of 2
|
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
|
Powers of 2: 2n for n ≥ 0
|
Template:OEIS link
|
| Polyominoes
|
1, 1, 1, 2, 5, 12, 35, 108, 369, ...
|
The number of free polyominoes with Template:Math cells.
|
Template:OEIS link
|
| Catalan numbers Template:Math
|
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...
|
|
Template:OEIS link
|
| Bell numbers Template:Math
|
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
|
Template:Math is the number of partitions of a set with Template:Math elements.
|
Template:OEIS link
|
| Euler zigzag numbers Template:Math
|
1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ...
|
Template:Math is the number of linear extensions of the "zig-zag" poset.
|
Template:OEIS link
|
| 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 Template:Math cuts.
|
Template:OEIS link
|
| Pell numbers Template:Math
|
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...
|
Template:Math for Template:Math, with Template:Math.
|
Template:OEIS link
|
| Factorials Template:Math
|
1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...
|
for Template:Math, with Template:Math (empty product).
|
Template:OEIS link
|
| Derangements
|
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...
|
Number of permutations of n elements with no fixed points.
|
Template:OEIS link
|
| Divisor function Template:Math
|
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...
|
Template:Math is the sum of divisors of a positive integer Template:Math.
|
Template:OEIS link
|
| Fermat numbers Template:Math
|
3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...
|
Template:Math for Template:Math.
|
Template:OEIS link
|
| Polytrees
|
1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...
|
Number of oriented trees with n nodes.
|
Template:OEIS link
|
| Perfect numbers
|
6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...
|
Template:Math is equal to the sum Template:Math of the proper divisors of Template:Math.
|
Template:OEIS link
|
| Ramanujan tau function
|
1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ...
|
Values of the Ramanujan tau function, Template:Math at n = 1, 2, 3, ...
|
Template:OEIS link
|
| Landau's function
|
1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...
|
The largest order of permutation of Template:Math elements.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Padovan sequence
|
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ...
|
Template:Math for Template:Math, with Template:Math.
|
Template:OEIS link
|
| Euclid–Mullin sequence
|
2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...
|
Template:Math is smallest prime factor of Template:Math.
|
Template:OEIS link
|
| Lucky numbers
|
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...
|
A natural number in a set that is filtered by a sieve.
|
Template:OEIS link
|
| Prime powers
|
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...
|
Positive integer powers of prime numbers
|
Template:OEIS link
|
| Central binomial coefficients
|
1, 2, 6, 20, 70, 252, 924, ...
|
, numbers in the center of even rows of Pascal's triangle
|
Template:OEIS link
|
| Motzkin numbers
|
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...
|
The number of ways of drawing any number of nonintersecting chords joining Template:Math (labeled) points on a circle.
|
Template:OEIS link
|
| Jordan–Pólya numbers
|
1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ...
|
Numbers that are the product of factorials.
|
Template:OEIS link
|
| Jacobsthal numbers
|
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...
|
Template:Math for Template:Math, with Template:Math.
|
Template:OEIS link
|
| Sum of proper divisors Template:Math
|
0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ...
|
Template:Math is the sum of the proper divisors of the positive integer Template:Math.
|
Template:OEIS link
|
| 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 Template:Math endpoints (and Template:Math nodes in all).
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Semiprimes
|
4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...
|
Products of two primes, not necessarily distinct.
|
Template:OEIS link
|
| Golomb sequence
|
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ...
|
Template:Math is the number of times Template:Math occurs, starting with Template:Math.
|
Template:OEIS link
|
| Perrin numbers Template:Math
|
3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ...
|
Template:Math for Template:Math, with Template:Math.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Cullen numbers Template:Math
|
1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...
|
Template:Math, with Template:Math.
|
Template:OEIS link
|
| Primorials Template:Math
|
1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...
|
Template:Math, the product of the first Template:Math primes.
|
Template:OEIS link
|
| Highly composite numbers
|
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...
|
A positive integer with more divisors than any smaller positive integer.
|
Template:OEIS link
|
| Superior highly composite numbers
|
2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...
|
A positive integer Template:Math for which there is an Template:Math such that Template:Math for all Template:Math.
|
Template:OEIS link
|
| Pronic numbers
|
0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...
|
Template:Math, with Template:Math where Template:Math are the triangular numbers.
|
Template:OEIS link
|
| Markov numbers
|
1, 2, 5, 13, 29, 34, 89, 169, 194, ...
|
Positive integer solutions of Template:Math.
|
Template:OEIS link
|
| Composite numbers
|
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...
|
The numbers Template:Math of the form Template:Math for Template:Math and Template:Math.
|
Template:OEIS link
|
| Ulam number
|
1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
|
Template:Math for Template:Math is least number Template:Math which is a unique sum of two distinct earlier terms; semiperfect.
|
Template:OEIS link
|
| Prime knots
|
0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...
|
The number of prime knots with n crossings.
|
Template:OEIS link
|
| Carmichael numbers
|
561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ...
|
Composite numbers Template:Math such that Template:Math if Template:Math is coprime with Template:Math.
|
Template:OEIS link
|
| Woodall numbers
|
1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...
|
Template:Math, with Template:Math.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Colossally abundant numbers
|
2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...
|
A number n is colossally abundant if there is an Template:Math such that for all Template:Math,
where Template:Mvar denotes the sum-of-divisors function.
|
Template:OEIS link
|
| 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 Template:Math.
|
Template:OEIS link
|
| Deficient numbers
|
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...
|
Positive integers Template:Math such that Template:Math.
|
Template:OEIS link
|
| Abundant numbers
|
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...
|
Positive integers Template:Math such that Template:Math.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Look-and-say sequence
|
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...
|
A = 'frequency' followed by 'digit'-indication.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Alternating factorial
|
1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...
|
|
Template:OEIS link
|
| Fortunate numbers
|
3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ...
|
The smallest integer Template:Math such that Template:Math is a prime number, where the primorial Template:Math is the product of the first Template:Math prime numbers.
|
Template:OEIS link
|
| Semiperfect numbers
|
6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ...
|
A natural number Template:Math that is equal to the sum of all or some of its proper divisors.
|
Template:OEIS link
|
| 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 Template:Math.
|
Template:OEIS link
|
| Weird numbers
|
70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...
|
A natural number that is abundant but not semiperfect.
|
Template:OEIS link
|
| Farey sequence numerators
|
0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...
|
|
Template:OEIS link
|
| Farey sequence denominators
|
1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...
|
|
Template:OEIS link
|
| Euclid numbers
|
2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ...
|
Template:Math, i.e. Template:Math product of first Template:Math consecutive primes.
|
Template:OEIS link
|
| Kaprekar numbers
|
1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...
|
Template:Math, where Template:Math and Template:Math.
|
Template:OEIS link
|
| Sphenic numbers
|
30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...
|
Products of 3 distinct primes.
|
Template:OEIS link
|
| Giuga numbers
|
30, 858, 1722, 66198, 2214408306, ...
|
Composite numbers so that for each of its distinct prime factors pi we have .
|
Template:OEIS link
|
| Radical of an integer
|
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...
|
The radical of a positive integer Template:Math is the product of the distinct prime numbers dividing Template:Math.
|
Template:OEIS link
|
| Thue–Morse sequence
|
0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...
|
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Blum integers
|
21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ...
|
Numbers of the form Template:Math where Template:Mvar and Template:Mvar are distinct primes congruent to Template:Math.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Superperfect numbers
|
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...
|
Positive integers Template:Math for which Template:Math
|
Template:OEIS link
|
| Bernoulli numbers Template:Math
|
1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...
|
|
Template:OEIS link
|
| Hyperperfect numbers
|
6, 21, 28, 301, 325, 496, 697, ...
|
Template:Math-hyperperfect numbers, i.e. Template:Math for which the equality Template:Math holds.
|
Template:OEIS link
|
| Achilles numbers
|
72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...
|
Positive integers which are powerful but imperfect.
|
Template:OEIS link
|
| Primary pseudoperfect numbers
|
2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...
|
Satisfies a certain Egyptian fraction.
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Sierpinski numbers
|
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...
|
Odd Template:Math for which Template:Math consists only of composite numbers.
|
Template:OEIS link
|
| Riesel numbers
|
509203, 762701, 777149, 790841, 992077, ...
|
Odd Template:Math for which Template:Math consists only of composite numbers.
|
Template:OEIS link
|
| Baum–Sweet sequence
|
1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ...
|
Template:Math if the binary representation of Template:Math contains no block of consecutive zeros of odd length; otherwise Template:Math.
|
Template:OEIS link
|
| Gijswijt's sequence
|
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ...
|
The Template:Mathth term counts the maximal number of repeated blocks at the end of the subsequence from Template:Math to Template:Math
|
Template:OEIS link
|
| Carol numbers
|
−1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...
|
|
Template:OEIS link
|
| Juggler sequence
|
0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...
|
If Template:Math then Template:Math else Template:Math.
|
Template:OEIS link
|
| Highly totient numbers
|
1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...
|
Each number Template:Mvar on this list has more solutions to the equation Template:Math than any preceding Template:Math.
|
Template:OEIS link
|
| Euler numbers
|
1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...
|
|
Template:OEIS link
|
| 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.
|
Template:OEIS link
|
| Erdős–Nicolas numbers
|
24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...
|
A number Template:Mvar such that there exists another number Template:Mvar and
|
Template:OEIS link
|
| Solution to Stepping Stone Puzzle
|
1, 16, 28, 38, 49, 60, ...
|
The maximal value Template:Math of the stepping stone puzzle
|
Template:OEIS link
|