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%I #28 Nov 13 2012 20:36:30
%S 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,1,0,0,0,1,1,1,0,1,1,2,1,2,0,2,
%T 1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,2,1,2,0,1,2,2,2,2,0,2,2,1,1,1,
%U 2,1,0,0,1,0,2,2,0,2,1,3,0,1,1,2,2,1,0,3,2,3,0,2,1,4,1,1,2,1,3,2
%N Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q<n/2 if n=4 (mod 6).
%C Conjecture: a(n)>0 for all n>50000 with n different from 50627, 61127, 66503.
%C This conjecture implies that there are infinitely many cousin prime pairs. It is similar to the conjectures related to A219157 and A219055.
%H Zhi-Wei Sun, <a href="/A218867/b218867.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%H Zhi-Wei Sun, <a href="/A218867/a218867.txt">Table of n, a(n) for n = 1..10^5</a>.
%e a(20)=1 since 20=11+3*3 with 11-4 and 3+4 prime. a(28)=1 since 28=41-13 with 41-4 and 13+4 prime.
%t c[n_]:=c[n]=If[Mod[n+2,6]==0,1,-1-Mod[n,6]]; d[n_]:=d[n]=2+If[Mod[n+2,6]>0,Mod[n,6],0]; a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+4] == True && PrimeQ[n+c[n]Prime[k]] == True && PrimeQ[n+c[n]Prime[k]-4]==True,1,0], {k,1,PrimePi[(n-1)/d[n]]}]; Do[Print[n," ",a[n]], {n,100}]
%Y Cf. A023200, A046132, A219157, A219055, A002375, A046927, A218754, A218825, A219052.
%K nonn
%O 1,30
%A _Zhi-Wei Sun_, Nov 13 2012
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