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%I #72 Mar 07 2023 18:53:17
%S 2,4,10,14290,64390,74554,83464,93460,132304,238850,262630,277630,
%T 300206,352600,376190,404954,415180,610340,806180,984686,1025650,
%U 1047050,1106116,1382860,2014624,2440714,2525870,2538344,2760026,2826380,3145600,3508586,3715156
%N Numbers k such that k^2 + {1,3,7,13,163} are prime.
%C For 14 <= m <= 999 and k <= A356110(31) = 8069560, the number of sets of primes of the form k^2 + {1,3,7,13,m} is the greatest for m = 163. There are 51 such terms. See b-file.
%C All terms are 2 or 4 modulo 6.
%H David A. Corneth, <a href="/A356175/b356175.txt">Table of n, a(n) for n = 1..6757</a> (first 237 terms from Jean-Marc Rebert, terms <= 10^10).
%e 2 is a term since 2^2 + {1,3,7,13,163} = {5,7,11,17,167} are all primes.
%p q:= k-> andmap(j-> isprime(k^2+j), [1, 3, 7, 13, 163]):
%p select(q, [$0..1000000])[]; # _Alois P. Heinz_, Jul 28 2022
%t Select[Range[4*10^6], AllTrue[#^2 + {1, 3, 7, 13, 163}, PrimeQ] &] (* _Amiram Eldar_, Jul 28 2022 *)
%o (PARI)
%o is(k)=my(v=[1,3,7,13,163],ok=1);for(i=1,#v,if(!isprime(k^2+v[i]),ok=0;break));ok
%o (Python)
%o from sympy import isprime
%o def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,163})
%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Jul 28 2022
%Y Cf. A356110.
%Y Subsequence of A005574, A049422, A114270, A113536, A080149, A182238 and A356109.
%K nonn
%O 1,1
%A _Jean-Marc Rebert_, Jul 28 2022
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