%I #18 Nov 30 2012 12:21:42

%S 0,0,1,1,2,3,0,2,4,2,2,4,1,2,6,3,1,2,2,5,3,1,1,7,2,6,3,1,6,8,2,2,5,3,

%T 3,8,2,4,6,3,4,4,1,3,7,2,3,7,3,6,8,2,1,12,5,4,7,4,7,7,7,5,4,4,6,9,2,2,

%U 13,2,5,7,2,4,18,6,3,5,6,5,8,4,2,9,4,10,5,2,5,17,3,3,7,7,5,8,3,3,17,8

%N Number of ways to write n as p+q with p and 2pq+1 both prime

%C Conjecture: a(n)>0 for all n>7.

%C This has been verified for n up to 3*10^8.

%C Zhi-Wei Sun also made the following general conjecture: For each odd integer m not congruent to 5 modulo 6, any sufficiently large integer n can be written as p+q with p and 2*p*q+m both prime.

%C For example, when m = 3, -3, 7, 9, -9, -11, 13, 15, it suffices to require that n is greater than 1, 29, 16, 224, 29, 5, 10, 52 respectively.

%C Sun also guessed that any integer n>4190 can be written as p+q with p, 2*p*q+1, 2*p*q+7 all prime, and any even number n>1558 can be written as p+q with p, q, 2*p*q+3 all prime. He has some other similar observations.

%H Zhi-Wei Sun, <a href="/A219864/b219864.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(10)=2 since 10=3+7=7+3 with 2*3*7+1=43 prime.

%e a(263)=1 since 83 is the only prime p with 2p(263-p)+1 prime.

%t a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k](n-Prime[k])+1]==True,1,0],{k,1,PrimePi[n]}]

%t Do[Print[n," ",a[n]],{n,1,1000}]

%Y Cf. A219842, A002372, A046927, A219838, A219791, A219782, A036468.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Nov 30 2012