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A341210
Primes p such that (p^16 + 1)/2 is prime.
7
3, 29, 41, 73, 113, 157, 167, 173, 199, 599, 607, 617, 1213, 1747, 1979, 2027, 2237, 2377, 2441, 2593, 2659, 2689, 2693, 3061, 3137, 3413, 3457, 3539, 3673, 3733, 3769, 4091, 4157, 4273, 4289, 4547, 4603, 4759, 4877, 4909, 4957, 5039, 5231, 5233, 5303, 5419
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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, and j=2^4=16, respectively.
LINKS
EXAMPLE
(3^16 + 1)/2 = 21523361 is prime, so 3 is a term.
(5^16 + 1)/2 = 76293945313 = 2593*29423041, so 5 is not a term.
MATHEMATICA
Select[Prime[Range[750]], PrimeQ[(#^16+1)/2]&] (* Harvey P. Dale, Oct 06 2023 *)
PROG
(PARI) isok(p) = isprime(p) && (p>2) && isprime((p^16 + 1)/2); \\ Michel Marcus, Feb 07 2021
CROSSREFS
Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), (this sequence) (k=4).
Sequence in context: A178642 A125854 A167278 * A106979 A367102 A087209
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Feb 06 2021
STATUS
approved