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%I #9 Jan 03 2013 23:09:23
%S 0,0,0,0,1,2,2,2,2,3,4,2,2,3,3,3,2,3,3,4,5,3,6,5,4,6,3,5,4,3,6,2,4,5,
%T 5,4,4,6,5,4,6,5,4,5,7,5,2,3,6,4,5,4,5,7,6,9,5,4,9,5,4,5,5,4,5,6,3,8,
%U 5,8,8,3,7,5,3,5,3,5,4,9,6,4,9,7,5,8,7,8,6,9,8,2,7,7,5,6,2,10,6,3
%N Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime
%C Conjecture: a(n)>0 for all n>4.
%C This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.
%H Zhi-Wei Sun, <a href="/A187758/b187758.txt">Table of n, a(n) for n = 1..20000</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(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+5 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Jan 03 2013
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