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Numbers k such that k^2 + {1,3,7,13,31} are prime.
1

%I #15 Jul 28 2022 03:52:10

%S 4,10,14290,43054,109456,315410,352600,483494,566296,685114,927070,

%T 1106116,1248796,1501174,1997986,2399204,2501404,2553100,2726840,

%U 2874680,3291760,4129394,4473766,4794520,4901144,6350306,7444070,7753456,7892504,8009536,8069540

%N Numbers k such that k^2 + {1,3,7,13,31} are prime.

%C Conjecture: the sequence is infinite.

%e 4^2 + {1,3,7,13,31} = {17,19,23,29,47} are all prime.

%p q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13,31]):

%p select(q, [$0..1000000])[]; # _Alois P. Heinz_, Jul 27 2022

%t Select[Range[10^6], AllTrue[#^2 + {1,3,7,13,31}, PrimeQ] &] (* _Amiram Eldar_, Jul 27 2022 *)

%o (Python)

%o from sympy import isprime

%o def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,31})

%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Jul 27 2022

%Y Cf. A005574, A049422, A114270, A113536, A182238, A356109.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jul 27 2022