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%I #9 Dec 02 2012 12:32:50
%S 0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,1,2,2,1,1,2,2,1,3,2,2,3,2,3,3,2,1,4,3,
%T 1,4,3,1,4,2,3,3,2,2,4,3,2,4,2,2,5,3,4,5,2,1,5,3,2,4,1,1,5,4,4,4,3,2,
%U 5,3,2,4,3,4,5,3,4,6,3,3,6,3,3,8,5,2,6,3,4,6,2,2,9,5,3,5,4,2,6,4
%N Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime
%C Conjecture: a(n)>0 for all n>11.
%C This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.
%H Zhi-Wei Sun, <a href="/A219966/b219966.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(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.
%e a(20)=1 since 20=13+7 with 13, 7, 7+6 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[n/2]}]
%t Do[Print[n," ",a[n]],{n,1,10000}]
%Y Cf. A219055, A023201, A002375, A046927.
%K nonn
%O 1,17
%A _Zhi-Wei Sun_, Dec 02 2012
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