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

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

Cf. A185636, A071558.

Sequence in context: A366766 A071960 A056898 * A275281 A204176 A062160

Adjacent sequences: A204062 A204063 A204064 * A204066 A204067 A204068

Keyword

nonn

Author

Zhi-Wei Sun, Jan 09 2013

Status

approved