A204065 - OEIS

A204065

Least nonnegative integer k with n+k and n+k^2 both prime.

1, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 16, 1, 0, 7, 4, 15, 2, 1, 0, 1, 0, 9, 8, 3, 6, 1, 0, 3, 2, 1, 0, 1, 0, 3, 8, 1, 0, 5, 10, 3, 10, 1, 0, 5, 4, 15, 2, 1, 0, 1, 0, 21, 4, 3, 6, 1, 0, 15, 2, 1, 0, 1, 0, 33, 8, 25, 6, 1, 0, 3, 16, 1, 0, 5, 4, 15, 14, 1, 0, 7, 6, 9, 4, 3, 6, 1, 0, 3, 2, 1

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OFFSET

1,8

COMMENTS

Conjecture: For any n > 0 not among 1, 21, 326, 341, 626, we have a(n) < sqrt(n)*log(n). If n > 626 is not equal to 971, then n+k and n+k^2 are both prime for some 0< k < sqrt(n)*log(n). Also, n+k^2 is prime for some 0< k <= sqrt(n) if n > 43181.

Obviously, a(n)=0 iff n is a prime. - M. F. Hasler, Jan 11 2013

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

EXAMPLE

a(8)=3 since 8+3 and 8+3^2 are both prime, but none of 8, 8+1, 8+2 is prime.

MATHEMATICA

Do[Do[If[PrimeQ[n+k]==True&&PrimeQ[n+k^2]==True, Print[n, " ", k]; Goto[aa]], {k, 0, n}];

Label[aa]; Continue, {n, 1, 100}]

PROG

(PARI) a(n)=my(k=0); while(!isprime(n+k) || !isprime(n+k^2), k++); k \\ - M. F. Hasler, Jan 11 2013

CROSSREFS