%I #13 Nov 29 2012 14:04:54
%S 0,1,1,1,1,3,1,1,4,2,2,4,2,2,7,4,1,3,4,5,5,4,2,8,2,7,4,2,8,11,5,3,8,7,
%T 5,14,7,5,10,8,7,8,4,8,9,5,4,11,6,11,14,5,3,19,12,7,11,6,9,12,13,8,9,
%U 10,12,16,5,6,22,8,11,11,5,10,26,15,5,11,15,10
%N Number of ways to write n as x+y (0<x<=y) with 2x*y+1 prime.
%C Conjecture: a(n)>0 for all n>1. Moreover, any integer n>357 can be written as x+y (x>0, y>0) with 2x*y+1 and 2x*y-1 twin primes.
%C This conjecture has been verified for n up to 10^8, and it implies the twin prime conjecture.
%C Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y, where x and y are positive integers with 2x*y+m and 2x*y-m both prime.
%C For example, when m=3,5,7,9,11 it suffices to require that n is greater than 5090, 222, 1785, 548, 603 respectively.
%H Zhi-Wei Sun, <a href="/A219842/b219842.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=1+9=3+7 with 2*1*9+1=19 and 2*3*7+1=43 both prime.
%t a[n_] := a[n] = Sum[If[PrimeQ[2k(n-k)+1] == True, 1, 0], {k, n/2}]; Do[Print[n," ", a[n]], {n, 100}]
%Y Cf. A091182, A219838, A219782.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Nov 29 2012