A390986 Integers k such that all prime factors of k^2+1 are Sophie Germain primes.

1, 2, 3, 7, 9, 12, 15, 17, 23, 32, 39, 40, 41, 45, 53, 77, 93, 110, 128, 144, 155, 160, 170, 171, 182, 198, 218, 219, 247, 250, 258, 301, 309, 318, 321, 322, 331, 332, 345, 347, 357, 373, 375, 383, 419, 426, 427, 440, 441, 442, 452, 479, 482, 490, 492, 500, 510

Offset

1,2

Comments

The sequence contains A389884.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

17 is a term because the prime factors of 17^2+1 = 290 are 2, 5 and 29 with 2 = A005384(1), 5 = A005384(3) and 29 = A005384(6).

Maple

with(numtheory):nn:=510:
for k from 1 to nn do:
 it:=0:d:=factorset(k^2+1):n0:=nops(d):
  for j from 1 to n0 do:
    if isprime(2*d[j]+1) then it:=it+1:else fi:od:
    if it=n0 then printf(`%d, `, k) else fi:od:
# Alternative:
q:= n-> andmap(p-> isprime(2*p+1), ifactors(n^2+1)[2][.., 1]):
select(q, [$1..510])[];  # Alois P. Heinz, Nov 25 2025

Mathematica

q[k_] := AllTrue[FactorInteger[k^2 + 1][[;; , 1]], PrimeQ[2*# + 1] &]; Select[Range[512], q] (* Amiram Eldar, Nov 25 2025 *)

Prog

(PARI) isok(k) = my(f=factor(k^2+1)[, 1]); #select(x->isprime(2*x+1), f) == #f; \\ Michel Marcus, Nov 25 2025

Crossrefs

Cf. A005384, A156541, A389884.

Sequence in context: A260641 A077510 A364052 * A178433 A073148 A014839

Adjacent sequences: A390983 A390984 A390985 * A390987 A390988 A390989

Keyword

nonn

Author

Michel Lagneau, Nov 25 2025

Status

approved