A341224 Primes p such that (p^32 + 1)/2 is prime.

3, 163, 181, 191, 229, 251, 839, 971, 1181, 1201, 1489, 1801, 1823, 1847, 1861, 1987, 2069, 2087, 3547, 3691, 3697, 6361, 6637, 6899, 6967, 7793, 7963, 8731, 8737, 10253, 10271, 10613, 10639, 10799, 11981, 12689, 12697, 13697, 13841, 14951, 15299, 16547, 16747

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, 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.

Links

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

Example

(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.

Maple

q:= p-> (q-> q(p) and q((p^32+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021

Mathematica

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

Prog

(PARI) isok(p) = (p>2) && isprime(p) && isprime((p^32 + 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), 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

Keyword

nonn

Author

Jon E. Schoenfield, Feb 07 2021

Status

approved