1,1

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, respectively.

(p^8 + 1)/2 is divisible by 17 when m mod 34 is 3, 5, 7, 11, 23, 27, 29, or 31.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

(3^8 + 1)/2 = 3281 = 17*193, so 3 is not a term.

(13^8 + 1)/2 = 407865361 is prime, so 13 is a term.

(17^8 + 1)/2 = 3487878721 = 18913 * 184417, so 17 is not a term.

Prime[Position[Table[(Prime[p]^8 + 1)/2, {p, 1, 803}], _Integer?PrimeQ]] // Flatten (* Robert P. P. McKone, Jan 31 2021 *)

(PARI) isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ Michel Marcus, Feb 01 2021

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), (this sequence) (k=3).

Sequence in context: A259427 A118857 A118529 * A129811 A108545 A136191

Adjacent sequences: A340477 A340478 A340479 * A340481 A340482 A340483

Jon E. Schoenfield, Jan 31 2021

approved