1,1

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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^5=32, respectively.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

(3^32 + 1)/2 = 926510094425921 is prime, so 3 is a term.

(5^32 + 1)/2 = 11641532182693481445313 = 641*75068993*241931001601, so 5 is not a term.

q:= p-> (q-> q(p) and q((p^32+1)/2))(isprime):

select(q, [$3..20000])[]; # Alois P. Heinz, Feb 07 2021

Select[Range[17000], PrimeQ[#] && PrimeQ[(#^32 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

(PARI) isok(p) = (p>2) && isprime(p) && isprime((p^32 + 1)/2); \\ Michel Marcus, Feb 07 2021

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), (this sequence) (k=5).

Cf. A000040.

Sequence in context: A157440 A157559 A157586 * A042439 A364302 A212845

Adjacent sequences: A341221 A341222 A341223 * A341225 A341226 A341227

Jon E. Schoenfield, Feb 07 2021

approved