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%I #14 Dec 22 2012 06:04:37
%S 0,0,1,2,2,2,2,3,2,3,0,4,2,4,3,2,2,3,3,5,3,3,3,4,4,3,2,4,3,5,3,4,3,5,
%T 5,6,3,4,3,5,5,5,6,5,4,5,5,6,7,5,4,5,4,7,6,4,4,4,5,6,6,5,6,7,4,5,2,4,
%U 7,5,7,4,5,6,7,7,7,5,6,4,7,4,7,7,6,5,3,5,8,7,7,5,5,6,4,5,4,5,8,7
%N Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime
%C Conjecture: a(n)>0 for all n>11.
%C This implies the twin prime conjecture, and it has been verified for n up to 10^9.
%C Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.
%H Zhi-Wei Sun, <a href="/A199800/b199800.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(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Dec 21 2012
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