%I #10 Feb 07 2021 22:53:34
%S 331,1783,2591,2791,7127,8443,9007,9859,10133,10883,10889,11621,12101,
%T 13183,15391,17737,19309,19571,21863,24043,24203,31159,32717,33377,
%U 34267,35023,35531,38177,39929,42397,43499,46867,49499,49943,50087,51137,53101,53377
%N Primes p such that (p^256 + 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, A340480, A341210, A341224, A341229, A341230, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^8=256, respectively.
%H Jon E. Schoenfield, <a href="/A341234/b341234.txt">Table of n, a(n) for n = 1..100</a>
%e (3^256 + 1)/2 = 6950422618...4449717761 (a 122-digit number) = 12289 * 8972801 * 891206124520373602817 * (a 90-digit prime), so 3 is not a term.
%e (331^256 + 1)/2 = 5955749334...7416010241 (a 645-digit number) is prime, so 331 is a term. Since 331 is the smallest prime p such that (p^256 + 1)/2 is prime, it is a(1) and is also A341211(8).
%t Select[Range[20000], PrimeQ[#] && PrimeQ[(#^256 + 1)/2] &] (* _Amiram Eldar_, Feb 07 2021 *)
%Y Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), A341230 (k=7), (this sequence) (k=8).
%Y Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Feb 07 2021