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%I #11 Jun 18 2022 15:59:46
%S 0,1,2,3,2,2,2,4,3,2,2,3,4,4,2,3,2,6,6,5,4,2,6,5,4,4,2,6,4,4,4,3,5,7,
%T 5,5,3,4,9,5,6,4,5,6,4,5,5,6,7,6,6,3,7,7,6,6,4,6,6,5,6,4,7,6,7,2,3,7,
%U 7,7,5,3,5,5,7,8,5,8,8,4,5,4,10,10,6,6,2,9,6,9,7,1,8,4,5,7,3,9,5,3
%N Number of ways to write n=x+y (x,y>0) with 6x-1, 6x+1, 6y+1 and 6y+5 all prime.
%C Conjecture: a(n)>0 for all n>1.
%C This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m-2 contains no prime for any integer m>1.
%H Zhi-Wei Sun, <a href="/A187757/b187757.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(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 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)+5]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A001359, A006512, A023200, A046132, A219157, A218867, A219185, A220455.
%K nonn,nice
%O 1,3
%A _Zhi-Wei Sun_, Jan 03 2013
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