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%I #9 Feb 09 2021 07:35:20
%S 827,10861,19501,22751,23339,23663,26347,29581,50077,62131,63331,
%T 70657,72221,73523,78301,85447,109013,122363,127363,149213,155461,
%U 170551,173549,183877,188579,206627,218149,220147,222029,226099,227231,232051,247601,248317,248543
%N Primes p such that (p^1024 + 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, A341264, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^10=1024, respectively.
%e (3^1024 + 1)/2 = 1866959243...6855178241 (a 489-digit number) = 59393 * 448524289 * 847036417 * 8273923970...2296603649 (a 466-digit composite number), so 3 is not a term.
%e (827^1024 + 1)/2 = 1677304013...0116613121 (a 2988-digit number) is prime, so 827 is a term. Since 827 is the smallest prime p such that (p^1024 + 1)/2 is prime, it is a(1) and is also A341211(10).
%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), A341264 (k=9), (this sequence) (k=10).
%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
%E a(17)-a(35) from _Jinyuan Wang_, Feb 09 2021