

A204065


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


7



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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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: A214845 A071960 A056898 * A275281 A204176 A062160
Adjacent sequences: A204062 A204063 A204064 * A204066 A204067 A204068


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 09 2013


STATUS

approved



