List of integer sequences

! Name !! First elements !! Short description !! OEIS

| 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...

| The {{math|n}}th term describes the length of the {{math|n}}th run

| {{OEIS link|A000002}}

| 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...

| {{math|φ(n)}} is the number of positive integers not greater than {{math|n}} that are coprime with {{math|n}}.

| {{OEIS link|A000010}}

| 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...

| {{math|L(n) {{=}} L(n − 1) + L(n − 2)}} for {{math|n ≥ 2}}, with {{math|L(0) {{=}} 2}} and {{math|L(1) {{=}} 1}}.

| {{OEIS link|A000032}}

| 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

| The prime numbers {{math|p_n}}, with {{math|n ≥ 1}}. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

| {{OEIS link|A000040}}

| 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...

| The partition numbers, number of additive breakdowns of n.

| {{OEIS link|A000041}}

| 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

| {{math|F(n) {{=}} F(n − 1) + F(n − 2)}} for {{math|n ≥ 2}}, with {{math|F(0) {{=}} 0}} and {{math|F(1) {{=}} 1}}.

| {{OEIS link|A000045}}

| 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...

| $a(n+1)=\; \backslash prod\_\{k=0\}^\{n\}\; a(k)+1=a(n)^2-a(n)+1$ for {{math|n ≥ 1}}, with {{math|a(0) {{=}} 2}}.

| {{OEIS link|A000058}}

| 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...

| {{math|T(n) {{=}} T(n − 1) + T(n − 2) + T(n − 3)}} for {{math|n ≥ 3}}, with {{math|T(0) {{=}} 0 and T(1) {{=}} T(2) {{=}} 1}}.

| {{OEIS link|A000073}}

| 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...

| Powers of 2: 2ⁿ for n ≥ 0

| {{OEIS link|A000079}}

| 1, 1, 1, 2, 5, 12, 35, 108, 369, ...

| The number of free polyominoes with {{math|n}} cells.

| {{OEIS link|A000105}}

| 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

| $C\_n\; =\; \backslash frac\{1\}\{n+1\}\{2n\backslash choose\; n\}\; =\; \backslash frac\{(2n)!\}\{(n+1)!\backslash ,n!\}\; =\; \backslash prod\backslash limits\_\{k=2\}^\{n\}\backslash frac\{n+k\}\{k\},\backslash quad\; n\; \backslash ge\; 0.$

| {{OEIS link|A000108}}

| 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...

| {{math|B_n}} is the number of partitions of a set with {{math|n}} elements.

| {{OEIS link|A000110}}

| 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ...

| {{math|E_n}} is the number of linear extensions of the "zig-zag" poset.

| {{OEIS link|A000111}}

| 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...

| The maximal number of pieces formed when slicing a pancake with {{math|n}} cuts.

| {{OEIS link|A000124}}

| 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...

| {{math|a(n) {{=}} 2a(n − 1) + a(n − 2)}} for {{math|n ≥ 2}}, with {{math|a(0) {{=}} 0, a(1) {{=}} 1}}.

| {{OEIS link|A000129}}

| 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...

| $n!\; =\backslash prod\_\{k=1\}^\{n\}\; k$ for {{math|n ≥ 1}}, with {{math|0! {{=}} 1}} (empty product).

| {{OEIS link|A000142}}

| 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...

| Number of permutations of n elements with no fixed points.

| {{OEIS link|A000166}}

| 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...

| {{math|σ(n) :{{=}} σ₁(n)}} is the sum of divisors of a positive integer {{math|n}}.

| {{OEIS link|A000203}}

| 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...

| {{math|F_n {{=}} 2²ⁿ + 1}} for {{math|n ≥ 0}}.

| {{OEIS link|A000215}}

| 1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...

| Number of oriented trees with n nodes.

| {{OEIS link|A000238}}

| 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...

| {{math|n}} is equal to the sum {{math|s(n) {{=}} σ(n) − n}} of the proper divisors of {{math|n}}.

| {{OEIS link|A000396}}

| 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ...

| Values of the Ramanujan tau function, {{math|τ(n)}} at n = 1, 2, 3, ...

| {{OEIS link|A000594}}

| 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...

| The largest order of permutation of {{math|n}} elements.

| {{OEIS link|A000793}}

| 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.

| {{OEIS link|A000930}}

| 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ...

| {{math|P(n) {{=}} P(n − 2) + P(n − 3)}} for {{math|n ≥ 3}}, with {{math|P(0) {{=}} P(1) {{=}} P(2) {{=}} 1}}.

| {{OEIS link|A000931}}

| 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...

| {{math|a(1) {{=}} 2; a(n + 1)}} is smallest prime factor of {{math|a(1) a(2) ⋯ a(n) + 1}}.

| {{OEIS link|A000945}}

| 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...

| A natural number in a set that is filtered by a sieve.

| {{OEIS link|A000959}}

| 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...

| Positive integer powers of prime numbers

| {{OEIS link|A000961}}

| 1, 2, 6, 20, 70, 252, 924, ...

| $\{2n\; \backslash choose\; n\}\; =\; \backslash frac\{(2n)!\}\{(n!)^2\}\backslash text\{\; for\; all\; \}n\; \backslash geq\; 0$, numbers in the center of even rows of Pascal's triangle

| {{OEIS link|A000984}}

| 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...

| The number of ways of drawing any number of nonintersecting chords joining {{math|n}} (labeled) points on a circle.

| {{OEIS link|A001006}}

| 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ...

| Numbers that are the product of factorials.

| {{OEIS link|A001013}}

| 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...

| {{math|a(n) {{=}} a(n − 1) + 2a(n − 2)}} for {{math|n ≥ 2}}, with {{math|a(0) {{=}} 0, a(1) {{=}} 1}}.

| {{OEIS link|A001045}}

| 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ...

| {{math|s(n) {{=}} σ(n) − n}} is the sum of the proper divisors of the positive integer {{math|n}}.

| {{OEIS link|A001065}}

| 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 {{math|n}} endpoints (and {{math|2n − 1}} nodes in all).

| {{OEIS link|A001190}}

| 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.

| {{OEIS link|A001316}}

| 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...

| Products of two primes, not necessarily distinct.

| {{OEIS link|A001358}}

| 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ...

| {{math|a(n)}} is the number of times {{math|n}} occurs, starting with {{math|a(1) {{=}} 1}}.

| {{OEIS link|A001462}}

| 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ...

| {{math|P(n) {{=}} P(n − 2) + P(n − 3)}} for {{math|n ≥ 3}}, with {{math|P(0) {{=}} 3, P(1) {{=}} 0, P(2) {{=}} 2}}.

| {{OEIS link|A001608}}

| 0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, ...

| Used in the analysis of comparison sorts.

| {{OEIS link|A001855}}

| 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...

| {{math|C_n {{=}} n⋅2ⁿ + 1}}, with {{math|n ≥ 0}}.

| {{OEIS link|A002064}}

| 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...

| {{math|p_n#}}, the product of the first {{math|n}} primes.

| {{OEIS link|A002110}}

| 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...

| A positive integer with more divisors than any smaller positive integer.

| {{OEIS link|A002182}}

| 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...

| A positive integer {{math|n}} for which there is an {{math|e > 0}} such that {{math| {{sfrac|d(n)|n^e}} ≥ {{sfrac|d(k)|k^e}} }} for all {{math|k > 1}}.

| {{OEIS link|A002201}}

| 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...

| {{math|a(n) {{=}} 2t(n) {{=}} n(n + 1)}}, with {{math|n ≥ 0}} where {{math|t(n)}} are the triangular numbers.

| {{OEIS link|A002378}}

| 1, 2, 5, 13, 29, 34, 89, 169, 194, ...

| Positive integer solutions of {{math|x² + y² + z² {{=}} 3xyz}}.

| {{OEIS link|A002559}}

| 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...

| The numbers {{math|n}} of the form {{math|xy}} for {{math|x > 1}} and {{math|y > 1}}.

| {{OEIS link|A002808}}

| 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...

| {{math|a(1) {{=}} 1; a(2) {{=}} 2;}} for {{math|n > 2, a(n)}} is least number {{math| > a(n − 1)}} which is a unique sum of two distinct earlier terms; semiperfect.

| {{OEIS link|A002858}}

| 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...

| The number of prime knots with n crossings.

| {{OEIS link|A002863}}

| 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ...

| Composite numbers {{math|n}} such that {{math|a^{n − 1} ≡ 1 (mod n)}} if {{math|a}} is coprime with {{math|n}}.

| {{OEIS link|A002997}}

| 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...

| {{math|n⋅2ⁿ − 1}}, with {{math|n ≥ 1}}.

| {{OEIS link|A003261}}

| 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.

| {{OEIS link|A003601}}

| 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...

| A number n is colossally abundant if there is an {{math|ε > 0}} such that for all {{math|k > 1}},

:$\backslash frac\{\backslash sigma(n)\}\{n^\{1+\backslash varepsilon\}\}\backslash geq\backslash frac\{\backslash sigma(k)\}\{k^\{1+\backslash varepsilon\}\},$

where {{mvar|σ}} denotes the sum-of-divisors function.

| {{OEIS link|A004490}}

| 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 {{math|n}}.

| {{OEIS link|A005044}}

| 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...

| Positive integers {{math|n}} such that {{math|σ(n) < 2n}}.

| {{OEIS link|A005100}}

| 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...

| Positive integers {{math|n}} such that {{math|σ(n) > 2n}}.

| {{OEIS link|A005101}}

| 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.

| {{OEIS link|A005114}}

| 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.

| {{OEIS link|A005132}}

| 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...

| A = 'frequency' followed by 'digit'-indication.

| {{OEIS link|A005150}}

| 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.

| {{OEIS link|A005153}}

| 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...

| $\backslash sum\_\{k=0\}^\{n-1\}\; (-1)^k\; (n-k)!$

| {{OEIS link|A005165}}

| 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ...

| The smallest integer {{math|m > 1}} such that {{math|p_n# + m}} is a prime number, where the primorial {{math|p_n#}} is the product of the first {{math|n}} prime numbers.

| {{OEIS link|A005235}}

| 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ...

| A natural number {{math|n}} that is equal to the sum of all or some of its proper divisors.

| {{OEIS link|A005835}}

| 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 {{math|n ≥ 3}}.

| {{OEIS link|A006003}}

| 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...

| A natural number that is abundant but not semiperfect.

| {{OEIS link|A006037}}

| 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...

|  

| {{OEIS link|A006842}}

| 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...

|  

| {{OEIS link|A006843}}

| 2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ...

| {{math|p_n# + 1}}, i.e. {{math|1 +}} product of first {{math|n}} consecutive primes.

| {{OEIS link|A006862}}

| 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...

| {{math|X² {{=}} Abⁿ + B}}, where {{math|0 < B < bⁿ}} and {{math|X {{=}} A + B}}.

| {{OEIS link|A006886}}

| 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...

| Products of 3 distinct primes.

| {{OEIS link|A007304}}

| 30, 858, 1722, 66198, 2214408306, ...

|Composite numbers so that for each of its distinct prime factors p_i we have $p\_i^2\; \backslash ,|\backslash ,\; (n\; -\; p\_i)$.

| {{OEIS link|A007850}}

| 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...

| The radical of a positive integer {{math|n}} is the product of the distinct prime numbers dividing {{math|n}}.

| {{OEIS link|A007947}}

| 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...

|

| {{OEIS link|A010060}}

| 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.

| {{OEIS link|A014577}}

| 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ...

| Numbers of the form {{math|pq}} where {{mvar|p}} and {{mvar|q}} are distinct primes congruent to {{math|3 (mod 4)}}.

| {{OEIS link|A016105}}

| 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.

| {{OEIS link|A018226}}

| 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...

| Positive integers {{math|n}} for which {{math|σ²(n) {{=}} σ(σ(n)) {{=}} 2n.}}

| {{OEIS link|A019279}}

| 1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...

|  

| {{OEIS link|A027641}}

| 6, 21, 28, 301, 325, 496, 697, ...

| {{math|k}}-hyperperfect numbers, i.e. {{math|n}} for which the equality {{math|n {{=}} 1 + k (σ(n) − n − 1)}} holds.

| {{OEIS link|A034897}}

| 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...

| Positive integers which are powerful but imperfect.

| {{OEIS link|A052486}}

| 2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...

| Satisfies a certain Egyptian fraction.

| {{OEIS link|A054377}}

| 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.

| {{OEIS link|A059756}}

| 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...

| Odd {{math|k}} for which {{math| { k⋅2ⁿ + 1 : n ∈ $\backslash mathbb\{N\}$ } }} consists only of composite numbers.

| {{OEIS link|A076336}}

| 509203, 762701, 777149, 790841, 992077, ...

| Odd {{math|k}} for which {{math| { k⋅2ⁿ − 1 : n ∈ $\backslash mathbb\{N\}$ } }} consists only of composite numbers.

| {{OEIS link|A076337}}

| 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ...

| {{math|a(n) {{=}} 1}} if the binary representation of {{math|n}} contains no block of consecutive zeros of odd length; otherwise {{math|a(n) {{=}} 0}}.

| {{OEIS link|A086747}}

| 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ...

| The {{math|n}}th term counts the maximal number of repeated blocks at the end of the subsequence from {{math|1}} to {{math|n−1}}

| {{OEIS link|A090822}}

| −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...

| $a(n)\; =\; (2^n\; -\; 1)^2\; -\; 2.$

| {{OEIS link|A093112}}

| 0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...

| If {{math|n ≡ 0 (mod 2)}} then {{math|⌊{{sqrt|n}}⌋}} else {{math|⌊n^3/2⌋}}.

| {{OEIS link|A094683}}

| 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...

| Each number {{mvar|k}} on this list has more solutions to the equation {{math|φ(x) {{=}} k}} than any preceding {{math|k}}.

| {{OEIS link|A097942}}

| 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...

| $\backslash frac\{1\}\{\backslash cosh\; t\}\; =\; \backslash frac\{2\}\{e^\{t\}\; +\; e^\; \{-t\}\; \}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{E\_n\}\{n!\}\; \backslash cdot\; t^n.$

| {{OEIS link|A122045}}

| 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.

| {{OEIS link|A138591}}

| 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...

| A number {{mvar|n}} such that there exists another number {{mvar|m}} and $\backslash sum\_\{d\; \backslash mid\; n,\backslash \; d\; \backslash leq\; m\}\backslash !d\; =\; n.$

| {{OEIS link|A194472}}

| 1, 16, 28, 38, 49, 60, ...

| The maximal value {{math|a(n)}} of the stepping stone puzzle

| {{OEIS link|A337663}}

! Name !! First elements !! Short description !! OEIS

| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

| The natural numbers (positive integers) {{math|n ∈ $\backslash mathbb\{N\}$}}.

| {{OEIS link|A000027}}

| 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...

| {{math|t(n) {{=}} C(n + 1, 2) {{=}} {{sfrac|n(n + 1)|2}} {{=}} 1 + 2 + ... + n}} for {{math|n ≥ 1}}, with {{math|t(0) {{=}} 0}} (empty sum).

| {{OEIS link|A000217}}

| 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...

| {{math|n² {{=}} n × n}}

| {{OEIS link|A000290}}

| 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ...

| {{math|T(n)}} is the sum of the first {{math|n}} triangular numbers, with {{math|T(0) {{=}} 0}} (empty sum).

| {{OEIS link|A000292}}

| 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ...

| {{math| {{sfrac|n(n + 1)(2n + 1)|6}} }}: The number of stacked spheres in a pyramid with a square base.

| {{OEIS link|A000330}}

| 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...

| {{math|n³ {{=}} n × n × n}}

| {{OEIS link|A000578}}

| 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...

| {{math|size=120%|1=n⁵}}

| {{OEIS link|A000584}}

| 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...

| S_n = 6n(n − 1) + 1.

| {{OEIS link|A003154}}

| 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, ...

| Stella octangula numbers: {{math|n(2n² − 1)}}, with {{math|n ≥ 0}}.

| {{OEIS link|A007588}}

! Name !! First elements !! Short description !! OEIS

| 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...

| Primes {{math|p}} such that {{math|2^p − 1}} is prime.

| {{OEIS link|A000043}}

| 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ...

| {{math|2^p − 1}} is prime, where {{math|p}} is a prime.

| {{OEIS link|A000668}}

| 3, 11, 43, 683, 2731, 43691, ...

| A prime number p of the form $p=\{\{2^q+1\}\backslash over\; 3\}$ where q is an odd prime.

| {{OEIS link|A000979}}

| 1093, 3511

| | Primes $p$ satisfying {{math|2^p−1 ≡ 1 (mod p²)}}.

| {{OEIS link|A001220}}

| 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ...

| A prime number {{math|p}} such that {{math|2p + 1}} is also prime.

| {{OEIS link|A005384}}

| 5, 13, 563

| | Primes $p$ satisfying {{math|(p−1)! ≡ −1 (mod p²)}}.

| {{OEIS link|A007540}}

| 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, ...

| The numbers whose trajectory under iteration of sum of squares of digits map includes {{math|1}}.

| {{OEIS link|A007770}}

| 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).

| {{OEIS link|A088054}}

| 16843, 2124679

| Primes $p$ satisfying $\{2p-1\; \backslash choose\; p-1\}\; \backslash equiv\; 1\; \backslash pmod\{p^4\}$.

| {{OEIS link|A088164}}

| 2, 11, 17, 29, 41, 47, 59, 67, ...

| The {{math|n}}^th Ramanujan prime is the least integer {{math|R_n}} for which {{math|π(x) − π(x/2) ≥ n}}, for all {{math|x ≥ R_n}}.

| {{OEIS link|A104272}}

! 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.

| {{OEIS link|A005224}}

| 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.

| {{OEIS link|A002113}}

| 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ...

| The numbers for which every permutation of digits is a prime.

| {{OEIS link|A003459}}

| 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).

| {{OEIS link|A005349}}

| 1, 2, 145, 40585, ...

| A natural number that equals the sum of the factorials of its decimal digits.

| {{OEIS link|A014080}}

| 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ...

| The numbers which remain prime under cyclic shifts of digits.

| {{OEIS link|A016114}}

| 1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ...

| For {{math|n ≥ 2, a(n)}} is the prime that is finally reached when you start with {{math|n}}, concatenate its prime factors (A037276) and repeat until a prime is reached; {{math|a(n) {{=}} −1}} if no prime is ever reached.

| {{OEIS link|A037274}}

| 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ...

| A number that has the digit form {{math|ababab}}.

| {{OEIS link|A046075}}

| 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.

| {{OEIS link|A046758}}

| 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).

| {{OEIS link|A046760}}

| 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, ...

| Numbers containing the digits {{math|0–9}} such that each digit appears exactly once.

| {{OEIS link|A050278}}