cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A276261 Centered 21-gonal primes.

Original entry on oeis.org

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089
Offset: 1

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Author

Ilya Gutkovskiy, Aug 26 2016

Keywords

Comments

Primes of the form (21*k^2 + 21*k + 2)/2.
Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

Links

Crossrefs

Cf. A000040, A069178.
Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Programs

  • Mathematica
    Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]
  • PARI
    lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A383025 Centered pentagonal numbers that are deficient.

Original entry on oeis.org

1, 16, 31, 51, 76, 106, 141, 181, 226, 331, 391, 526, 601, 681, 766, 856, 951, 1051, 1156, 1381, 1501, 1756, 1891, 2031, 2326, 2481, 2641, 2806, 3151, 3331, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5641, 5881, 6376, 6631, 6891, 7156, 7426, 7701, 7981, 8266, 8851, 9151, 9766, 10081, 10401
Offset: 1

Views

Author

Massimo Kofler, Apr 13 2025

Keywords

Comments

The centered pentagonal numbers that are prime are terms (see A145838).

Examples

			16 = 2^4 is a term since it is the 3rd centered pentagonal number and larger than the sum of its proper divisors (1+2+4+8=15).
51 = 3*17 is a term since it is the 5th centered pentagonal number and larger than the sum of its proper divisors (1+3+17=21).
76 = 2^2*19 is a term since it is the 6th centered pentagonal number and larger than the sum of its proper divisors (1+2+4+19+38=64).

Links

Crossrefs

Intersection of A005891 and A005100.
Cf. A145838, A381488, A382696.

Programs

  • Maple
    select(t -> numtheory:-sigma(t) < 2*t, [seq( (5*n^2+5*n+2)/2, n=0..100)]); # Robert Israel, May 13 2025
  • Mathematica
    Select[Table[(5*n^2 + 5*n + 2)/2, {n, 0, 65}], DivisorSigma[-1, #] < 2 &] (* Amiram Eldar, Apr 13 2025 *)

Formula

a(n) == 1 (mod 5).

A145837 Indices of primes in A005891(n).

Original entry on oeis.org

3, 8, 11, 15, 20, 23, 36, 44, 47, 48, 60, 68, 71, 75, 83, 84, 87, 92, 111, 116, 128, 132, 143, 144, 156, 159, 164, 167, 168, 183, 192, 200, 204, 207, 215, 224, 228, 231, 236, 239, 264, 272, 287, 299, 300, 303, 312, 315, 320, 323, 356, 359, 360, 363, 372, 387
Offset: 1

Views

Author

Alexander Adamchuk, Mar 21 2009

Keywords

Comments

Corresponding centered pentagonal primes are listed in A145838 = {31, 181, 331, 601, 1051, 1381, 3331, ...}.

Links

Crossrefs

Cf. A145838 = Primes in A005891. Cf. A005891 = Centered pentagonal numbers: (5n^2+5n+2)/2.

Programs

Formula

A005891(a(n)) = A145838(n)

Extensions

Extended by R. J. Mathar, Mar 26 2009
Edited by N. J. A. Sloane, Apr 06 2009

A381043 Centered pentagonal numbers which are squarefree semiprimes.

Original entry on oeis.org

6, 51, 106, 141, 226, 391, 526, 681, 766, 951, 1501, 1891, 2031, 2326, 2481, 2641, 3151, 3901, 4101, 4306, 6631, 6891, 7981, 8266, 8851, 10081, 10401, 11391, 13141, 14631, 15406, 16201, 20931, 23281, 24751, 27301, 27826, 28891, 29431, 30526, 32206, 33351, 35701, 36301, 38131, 38751, 41926
Offset: 1

Views

Author

Massimo Kofler, Apr 14 2025

Keywords

Comments

Numbers such as 22801=151^2 and 1666681=1291^2 are in A382132 but not here.

Examples

			A005891(1) = 6 = (5*1^2 + 5*1 + 2)/2 = 2*3.
A005891(4) = 51 = (5*4^2 + 5*4 + 2)/2 = 3*17.
A005891(6) = 106 = (5*6^2 + 5*6 + 2)/2 = 2*53.

Crossrefs

Intersection of A006881 and A005891.
Cf. A145838, A382132, A364610.

Programs

  • Mathematica
    Select[Table[5*n*(n + 1)/2 + 1, {n, 0, 150}], FactorInteger[#][[;; , 2]] == {1, 1} &] (* Amiram Eldar, Apr 14 2025 *)
Showing 1-4 of 4 results.

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