A204065 - OEIS

%I #16 Jan 14 2013 02:05:14

%S 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,

%T 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,

%U 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

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

%C 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.

%C Obviously, a(n)=0 iff n is a prime. - _M. F. Hasler_, Jan 11 2013

%H Zhi-Wei Sun, <a href="/A204065/b204065.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.

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

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

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

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

%Y Cf. A185636, A071558.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Jan 09 2013