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%I #6 Feb 07 2021 19:21:08
%S 3631,5113,10651,12391,13999,22093,34687,38713,38959,39199,39679,
%T 44879,51229,57389,58757,59651,60331,61543,63389,64483,72931,77023,
%U 80369,91639,100787,115679,119551,120713,121727,122299,132109,135599,140221,143387,143873,145753
%N Primes p such that (p^512 + 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, A341234, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^9=512, respectively.
%e (3^512 + 1)/2 = 9661674916...6218270721 (a 244-digit number) = 134382593 * 22320686081 * 12079910333441 * 100512627347897906177 * 2652879528...2021744641 (a 193-digit composite number), so 3 is not a term.
%e (3631^512 + 1)/2 = 2706508826...0763924481 (an 1823-digit number) is prime, so 3631 is a term. Since 3631 is the smallest prime p such that (p^512 + 1)/2 is prime, it is a(1) and is also A341211(9).
%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), A341234 (k=8), (this sequence) (k=9).
%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
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