A390701 - OEIS

%I #15 Nov 16 2025 22:18:58

%S 1,2,6,26,36,54,94,116,224,384,396,496,584,644,864,1066,1314,1354,

%T 1456,1674,1676,1736,1884,1966,1974,2026,2094,2314,2326,2404,2456,

%U 2576,2646,2746,2794,2834,2926,3134,3174,3196,3504,3536,3686,3754,3764,3806,4006,4184

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

%H Paolo Xausa, <a href="/A390701/b390701.txt">Table of n, a(n) for n = 1..10000</a>

%e 1 is in the sequence because 1^2+1 = 2 and 4*2^2+1 = 17 are prime numbers,

%e 2 is in the sequence because 2^2+1 = 5 and 4*5^2+1 = 101 are prime numbers,

%e 6 is in the sequence because 6^2+1 = 37 and 4*37^2+1 = 5477 are prime numbers.

%p for k from 1 to 4500 do:

%p   p:=k^2+1:q:=4*p^2+1:

%p     if isprime(p) and isprime(q) then printf(`%d, `,k):

%p     else

%p    fi :

%p  od:

%t Sqrt[Select[Range[5000]^2 + 1, PrimeQ[#] && PrimeQ[4*#^2 + 1] &] - 1] (* _Paolo Xausa_, Nov 16 2025 *)

%o (Python)

%o from sympy import isprime

%o def A390701_isok(k): return isprime(p:=k**2+1) and isprime(4*p**2+1)

%o print([k for k in range(1,4200) if A390701_isok(k)]) # _Karl-Heinz Hofmann_, Nov 16 2025

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

%K nonn

%O 1,2

%A _Michel Lagneau_, Nov 15 2025