OFFSET
1,1
COMMENTS
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.
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..100
EXAMPLE
(3^256 + 1)/2 = 6950422618...4449717761 (a 122-digit number) = 12289 * 8972801 * 891206124520373602817 * (a 90-digit prime), so 3 is not a term.
(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).
MATHEMATICA
Select[Range[20000], PrimeQ[#] && PrimeQ[(#^256 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Feb 07 2021
STATUS
approved