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 1-3 of 3 results.

A378143 a(n) is the smallest prime of the form (2*p)^(2^n) + 1 for some prime p.

Original entry on oeis.org

5, 17, 257, 65537, 808551180810136214718004658177, 9807585394417153072393128067370344132933540474708183331242417216238928121991128579833857
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Nov 17 2024

Keywords

Comments

If p = 2, then a(n) is the Fermat prime.
Conjecture: the last digit of each value of a(n), where n >= 1, is 7.
The conjecture is equivalent to the claim that a(n) is not 10^(2^n) + 1 for any n, which in turn is equivalent to the claim that, if 10^(2^n) + 1 is prime, then either 4^(2^n) + 1 or 6^(2^n) + 1 is prime. - Charles R Greathouse IV, Nov 17 2024

Crossrefs

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).
Cf. A019434, A222008, A286678, A378134.

A378146 Primes p such that 16*p^4 + 1 is prime.

Original entry on oeis.org

2, 3, 17, 23, 37, 41, 53, 59, 71, 97, 127, 139, 167, 233, 263, 277, 283, 379, 389, 457, 521, 563, 571, 601, 619, 661, 691, 743, 797, 809, 811, 823, 853, 859, 877, 967, 971, 997, 1051, 1063, 1103, 1187, 1277, 1289, 1321, 1367, 1399, 1433, 1451, 1499
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 17 2024

Keywords

Crossrefs

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), this sequence (k = 2).
Cf. A378134, A378143.

Programs

  • Magma
    [p: p in PrimesUpTo(1500) | IsPrime(16*p^4+1)];
  • Mathematica
    Select[Prime[Range[250]], PrimeQ[16*#^4 + 1] &] (* Amiram Eldar, Nov 17 2024 *)
  • PARI
    list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(16*p^4+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Nov 17 2024

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Nov 17 2024

A378702 Primes p such that 256*p^8 + 1 is prime.

Original entry on oeis.org

2, 59, 271, 281, 433, 467, 587, 971, 1039, 1097, 1181, 1277, 1283, 1361, 1373, 1427, 1447, 1481, 1579, 1657, 1777, 2089, 2129, 2269, 2381, 2617, 2753, 2803, 2939, 3181, 3319, 3691, 3823, 4093, 4217, 4241, 4327, 4909, 4999, 5279, 5303, 5387, 5483, 6043, 6121, 6197, 6221, 6563, 6577, 7159, 7243, 7867
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 04 2024

Keywords

Crossrefs

Primes p such that (2*p)^(2^n) + 1 is prime: A005384 (n = 0), A052291 (n = 1), A378146 (n = 2), this sequence (n = 3).
Cf. A006314, A378134, A378143.

Programs

  • Magma
    [p: p in PrimesUpTo(8000) | IsPrime(256*p^8 + 1)];
  • Mathematica
    Select[Prime[Range[1000]], PrimeQ[(2*#)^8 + 1] &] (* Amiram Eldar, Dec 06 2024 *)
  • PARI
    select(p->isprime(256*p^8+1), primes(10^6)) \\ Charles R Greathouse IV, Dec 04 2024

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 04 2024
Showing 1-3 of 3 results.

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

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