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%I #18 Feb 28 2013 13:14:31
%S 0,0,1,1,1,1,1,2,1,1,1,2,2,1,2,0,2,1,3,2,1,2,1,2,2,2,2,3,1,3,1,2,3,2,
%T 6,1,3,1,2,4,3,4,4,1,3,1,3,5,2,6,1,3,2,2,5,2,5,2,3,1,2,3,5,2,4,0,0,3,
%U 1,6,2,3,3,1,5,1,5,3,3,3,1,4,2,3,3,0,3,3,3,4,1,3,1,2,3,2,4,2,2,3
%N Number of ways to write n=x+y (0<x<y<n) with 6x-1, 6x+1, 6y-1 and 6y+1 all prime.
%C Conjecture: If n>200 is not among 211, 226, 541, 701, then a(n)>0.
%C This essentially follows from the conjecture related to A219157, since n=x+y for some positive integers x and y with 6x-1,6x+1,6y-1,6y+1 all prime if and only if 6n=p+q for some twin prime pairs {p,p-2} and {q,q+2}.
%C Similarly, the conjecture related to A218867 implies that any integer n>491 can be written as x+y (0<x<=y<n) with 6x+1, 6x+5, 6y+1 and 6y+5 all prime; and the conjecture related to A219055 implies that any integer n>1600 not among 2729 and 4006 can be written as x+y (0<x<=y<n) with 2x-3, 2x+3, 2y-3 and 2y+3 all prime.
%H Zhi-Wei Sun, <a href="/A187759/b187759.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(9)=1 since 9=2+7 with 6*2-1, 6*2+1, 6*7-1 and 6*7+1 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)-1]==True&&PrimeQ[6(n-k)+1]==True,1,0],{k,1,(n-1)/2}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%o (PARI) a(n)=sum(x=1,(n-1)\2,isprime(6*x-1)&&isprime(6*x+1)&&isprime(6*n-6*x-1)&&isprime(6*n-6*x+1)) \\ _Charles R Greathouse IV_, Feb 28 2013
%Y Cf. A001359, A006512, A219157, A219185, A199920, A187757, A187758, A218867, A219055.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Jan 03 2013
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