A378146 -id:A378146 - cp's OEIS frontend

A378134 a(n) is the smallest prime p such that (2*p)^(2^n) + 1 is also prime.

2, 2, 2, 2, 37, 281, 137, 2129, 139, 23, 1231, 1279, 17477

Offset: 0

Juri-Stepan Gerasimov, Nov 17 2024

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).
If a(n) is the smallest prime number p such that (p*2^m)^(2^n) + 1, then we have:
2, 2, 2, 2, 2 (in case m = 0), where primes of the form (p*2^0)^(2^n)+1 are A019434;
this sequence (in case m = 1).
Cf. A378143.

a(11)-a(12) from Michael S. Branicky, Nov 18 2024

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

Juri-Stepan Gerasimov, Nov 17 2024

nonn

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

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.

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

Juri-Stepan Gerasimov, Dec 04 2024

nonn

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.

[p: p in PrimesUpTo(8000) | IsPrime(256*p^8 + 1)];

Select[Prime[Range[1000]], PrimeQ[(2*#)^8 + 1] &] (* Amiram Eldar, Dec 06 2024 *)

select(p->isprime(256*p^8+1), primes(10^6)) \\ Charles R Greathouse IV, Dec 04 2024

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 04 2024

cp's OEIS Frontend

A378134 a(n) is the smallest prime p such that (2*p)^(2^n) + 1 is also prime.

Views

Author

Keywords

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Formula