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.

A178336 Smaller member of a twin prime pair of the form (k^3 + 2, k^3 + 4).

Original entry on oeis.org

3, 29, 91127, 250049, 328511, 2146691, 47832149, 121287377, 170953877, 194104541, 693154127, 979146659, 1167575879, 1664006627, 5079577961, 6219352721, 8678316377, 10289109377, 10633486601, 13980103931, 17474794877, 28066748321, 28736971049
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

Keywords

Examples

			3 = 1^3+2 = prime(2) and 5 = 1^3+4 = prime(3) are a twin prime pair, so 3 becomes the first term.
91127 = 45^3+2 = prime(8811) and 91129 = 45^3+4 = prime(8812) are a twin prime pair, so 91127 is a term.

References

  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909

Links

Crossrefs

Cf. A013159, A053703, A132282, A144953, A173255, A178228.

Programs

  • Mathematica
    Select[Range[3100]^3+2,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, May 26 2012 *)

Formula

a(n) = A178337(n)^3 + 2.

Extensions

Keyword:base removed, 2 missing terms inserted by R. J. Mathar, Jun 27 2010

A178337 Numbers k such that (k^3 + 2, n^3 + 4) is a twin prime pair.

Original entry on oeis.org

1, 3, 45, 63, 69, 129, 363, 495, 555, 579, 885, 993, 1053, 1185, 1719, 1839, 2055, 2175, 2199, 2409, 2595, 3039, 3063, 3303, 3399, 3555, 3615, 4245, 4443, 4449, 5073, 5373, 5535, 5703, 5949, 6015, 6075, 6693, 6795, 6849, 7023, 7119, 7155, 7509, 7779, 8535
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

Keywords

Comments

With the exception of k = 1, all k are odd multiples of 3 with a least-significant decimal digit of 3, 5 or 9.
A178336(n) gives the values of k^3 + 2.

Examples

			1^3 + 2 = 3 = prime(2) and 3+2 = prime(3) are twin primes, so n=1 is a term.
45^3 + 2 = 91127 = prime(8811) and 91127+2 = prime(8812) are twin primes, so 45 is a term.
10893^3 + 2 = 1292535591959 = prime(48144179941) is a lower twin prime, so 10893 is a term.

Links

Crossrefs

Cf. A013159, A053703, A132282, A144953, A173255, A178336.

Programs

  • Magma
    [n: n in [0..9000] | IsPrime(n^3+2) and IsPrime(n^3+4)]; // Vincenzo Librandi, Nov 18 2010
  • Mathematica
    seqQ[n_] := And @@ PrimeQ[n^3 + 3 + {-1, 1}]; Select[Range[8535], seqQ] (* Amiram Eldar, Jan 11 2020*)

Extensions

Keyword:base removed by R. J. Mathar, Jun 27 2010

A115232 Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 17 2006

Keywords

Comments

a(n)=A000040(n+2) for n <= 32, but A000040(35)=149 is a term of A115231;
A115233 is a subsequence; the union with A115231 gives all primes (A000040);
A006512 and A053703 are subsequences.

Crossrefs

Cf. A000079, A061345.
Cf. A006512, A053703, A115231, A115233.

Programs

  • Mathematica
    maxp = 281;
Union[Sort[Reap[Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], Sow[p]], {i, 0, Log[2, maxp]//Ceiling}, {j, 1, Log[3, maxp]//Ceiling}, {q, Prime[Range[2, PrimePi[maxp]]]}]][[2, 1]]]] (* Jean-François Alcover, Aug 03 2018 *)

Extensions

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010
Edited by N. J. A. Sloane, Apr 30 2010
Terms a(1)=2 and a(2)=3 removed from Data by Jean-François Alcover, Aug 03 2018

A178506 Lesser of a "near cube" twin prime pair (k^3 - 4, k^3 - 2).

Original entry on oeis.org

3371, 8120597, 69426527, 108531329, 176558477, 1207949621, 2379270371, 3477265871, 3560550179, 4227952109, 8012005997, 12665630687, 13060888871, 15832158827, 15945922409, 18337088849, 20279414579, 22354272509, 30283802609, 60559558979, 70496180087, 98035951127
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010

Keywords

Comments

p + 2 = k^3 - 2 is form of "near(est) cube" prime smaller than cube number k^3, as k^3 - 1 = (k-1) * (k^2 + k + 1), only prime for k=2.

Examples

			p = 3371 = prime(475) = 15^3 - 4, (p, p+2) is twin prime pair tp(90), 3371 is the first term.

Links

Crossrefs

Cf. A178336.
Cf. A001359, A053703, A132282, A144953, A173255, A178228, A038600.

Programs

  • Mathematica
    Select[Range[10^4]^3 - 4, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 25 2019 *)

Extensions

a(13) corrected and more terms from Amiram Eldar, Dec 25 2019
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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