A340480 - OEIS

%I #16 Feb 06 2021 22:08:56

%S 13,43,47,53,239,373,409,433,491,557,577,859,1021,1103,1307,1531,1699,

%T 1753,1777,1871,2053,2083,2297,2467,2503,2593,2797,2957,3251,3307,

%U 3323,3511,3613,4099,4523,4637,4951,4999,5591,5657,5693,5801,5827,5849,6043,6163

%N Primes p such that (p^8 + 1)/2 is prime.

%C 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.

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

%H Jon E. Schoenfield, <a href="/A340480/b340480.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

%o (PARI) isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ _Michel Marcus_, Feb 01 2021

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

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Jan 31 2021