A390701 Integers k such that p = k^2+1 and 4*p^2+1 are prime numbers.

1, 2, 6, 26, 36, 54, 94, 116, 224, 384, 396, 496, 584, 644, 864, 1066, 1314, 1354, 1456, 1674, 1676, 1736, 1884, 1966, 1974, 2026, 2094, 2314, 2326, 2404, 2456, 2576, 2646, 2746, 2794, 2834, 2926, 3134, 3174, 3196, 3504, 3536, 3686, 3754, 3764, 3806, 4006, 4184

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

1 is in the sequence because 1^2+1 = 2 and 4*2^2+1 = 17 are prime numbers,
2 is in the sequence because 2^2+1 = 5 and 4*5^2+1 = 101 are prime numbers,
6 is in the sequence because 6^2+1 = 37 and 4*37^2+1 = 5477 are prime numbers.

Maple

for k from 1 to 4500 do:
  p:=k^2+1:q:=4*p^2+1:
    if isprime(p) and isprime(q) then printf(`%d, `, k):
    else
   fi :
 od:

Mathematica

Sqrt[Select[Range[5000]^2 + 1, PrimeQ[#] && PrimeQ[4*#^2 + 1] &] - 1] (* Paolo Xausa, Nov 16 2025 *)

Prog

(Python)
from sympy import isprime
def A390701_isok(k): return isprime(p:=k**2+1) and isprime(4*p**2+1)
print([k for k in range(1, 4200) if A390701_isok(k)]) # Karl-Heinz Hofmann, Nov 16 2025

Crossrefs

Cf. A002496, A001912, A005574, A052291, A052292.

Sequence in context: A029988 A050573 A337573 * A331963 A308988 A343753

Adjacent sequences: A390698 A390699 A390700 * A390702 A390703 A390704

Keyword

nonn

Author

Michel Lagneau, Nov 15 2025

Status

approved