A356175 Numbers k such that k^2 + {1,3,7,13,163} are prime.

2, 4, 10, 14290, 64390, 74554, 83464, 93460, 132304, 238850, 262630, 277630, 300206, 352600, 376190, 404954, 415180, 610340, 806180, 984686, 1025650, 1047050, 1106116, 1382860, 2014624, 2440714, 2525870, 2538344, 2760026, 2826380, 3145600, 3508586, 3715156

Offset

1,1

Comments

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.

All terms are 2 or 4 modulo 6.

Links

David A. Corneth, Table of n, a(n) for n = 1..6757 (first 237 terms from Jean-Marc Rebert, terms <= 10^10).

Example

2 is a term since 2^2 + {1,3,7,13,163} = {5,7,11,17,167} are all primes.

Maple

q:= k-> andmap(j-> isprime(k^2+j), [1, 3, 7, 13, 163]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 28 2022

Mathematica

Select[Range[4*10^6], AllTrue[#^2 + {1, 3, 7, 13, 163}, PrimeQ] &] (* Amiram Eldar, Jul 28 2022 *)

Prog

(PARI)
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
(Python)
from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1, 3, 7, 13, 163})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 28 2022

Crossrefs

Cf. A356110.

Subsequence of A005574, A049422, A114270, A113536, A080149, A182238 and A356109.

Sequence in context: A012616 A012611 A356109 * A133757 A246164 A124183

Adjacent sequences: A356172 A356173 A356174 * A356176 A356177 A356178

Keyword

nonn

Author

Jean-Marc Rebert, Jul 28 2022

Status

approved