%I
%S 0,0,1,0,1,0,1,2,1,1,1,0,3,2,1,1,4,3,1,1,3,3,2,3,3,1,2,3,4,2,1,6,4,4,
%T 1,4,2,1,5,4,2,1,2,4,2,2,3,3,3,4,2,3,3,2,3,1,5,2,3,1,5,6,4,5,3,3,1,4,
%U 3,2,3,5,3,3,7,4,3,1,4,5,4,3,2,4,2,5,5,4,2,2,6,8,2,2,4,2,6,1,3,2
%N Number of ordered ways to write n = p + q with q > 0 such that p, 3*(p + prime(q))  1 and 3*(p + prime(q)) + 1 are all prime.
%C Conjecture: a(n) > 0 if n is not a divisor of 12.
%C Clearly, this implies the twin prime conjecture.
%H ZhiWei Sun, <a href="/A182662/b182662.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(11) = 1 since 11 = 7 + 4 with 7, 3*(7 + prime(4))  1 = 3*14  1 = 41 and 3*(7 + prime(4)) + 1 = 3*14 + 1 = 43 all prime.
%e a(210) = 1 since 210 = 97 + 113 with 97, 3*(97 + prime(113))  1 = 3*(97 + 617)  1 = 2141 and 3*(97 + prime(113)) + 1 = 3*(97 + 617) + 1 = 2143 all prime.
%t p[n_,m_]:=PrimeQ[3(m+Prime[nm])1]&&PrimeQ[3(m+Prime[nm])+1]
%t a[n_]:=Sum[If[p[n,Prime[k]],1,0],{k,1,PrimePi[n1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A001359, A006512, A236531, A236831.
%K nonn
%O 1,8
%A _ZhiWei Sun_, Jan 31 2014
