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%I #6 Nov 12 2012 22:18:58
%S 0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,1,1,0,0,0,2,1,2,1,1,1,2,1,1,1,1,1,1,
%T 1,2,2,1,1,1,2,1,1,1,0,2,0,2,2,1,1,2,3,1,0,2,1,1,0,2,2,1,2,1,2,1,0,1,
%U 0,2,1,2,1,1,1,2,1,2,2,1,1,2,2,2,1,1,2,1,1,2,2,2,1,2,2,0,1,3,1,0
%N Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q<n/2 if n=2 (mod 6).
%C Conjecture: a(n)>0 for all n>30000 with n different from 38451, 46441, 50671, 62371.
%C This conjecture is stronger than the twin prime conjecture. It is similar to the conjecture associated with A219055 about sexy prime pairs.
%H Zhi-Wei Sun, <a href="/A219157/b219157.txt">Table of n, a(n) for n = 1..100000</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(16)=1 since 16=7+3*3 with 7-2 and 3+2 prime. a(26)=1 since 26=31-5 with 31-2 and 5+2 prime.
%t c[n_]:=c[n]=If[Mod[n-2,6]==0,1,-1-Mod[-n,6]]
%t d[n_]:=d[n]=2+If[Mod[n-2,6]>0,Mod[-n,6],0]
%t a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[n+c[n]Prime[k]]==True&&PrimeQ[n+c[n]Prime[k]-2]==True,1,0],
%t {k,1,PrimePi[(n-1)/d[n]]}]
%t Do[Print[n," ",a[n]],{n,1,100000}]
%Y Cf. A001359, A006512, A002375, A046927, A219055, A218754, A218825, A219052.
%K nonn
%O 1,22
%A _Zhi-Wei Sun_, Nov 12 2012
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