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%I #6 Dec 01 2012 13:51:34
%S 0,0,0,0,0,1,1,0,2,0,1,1,1,1,3,2,0,1,2,2,3,2,1,0,2,2,0,1,3,2,2,1,3,4,
%T 2,2,3,0,4,3,3,1,1,3,0,3,2,1,1,3,3,1,1,5,3,1,2,1,3,3,5,3,1,2,4,3,3,2,
%U 4,3,2,2,0,3,5,4,1,3,6,2,6,2,2,4,5,5,2,3,3,4,1,2,0,1,4,2,4,1,6,6
%N Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2*x*y+1 all prime
%C Conjecture: a(n)>0 for all n>623.
%C This has been verified for n up to 10^8.
%C Zhi-Wei Sun made the following general conjecture: For each nonnegative integer m, any sufficiently large integer n can be written as x+y (x>0, y>0) with x-m, x+m and 2*x*y+1 all prime.
%C For example, when m = 2, 3, 4, 5 it suffices to require that n is greater than 28, 151, 357, 199 respectively.
%C Sun also conjectured that for each m=0,1,2,... any sufficiently large integer n with m or n odd can be written as x+y (x>0, y>0) with x-m, x+m and x*y-1 all prime.
%C For example, in the case m=1 it suffices to require that n is greater than 4 and not among 40, 125, 155, 180, 470, 1275, 2185, 3875; when m=2 it suffices to require that n is odd, greater than 7, and different from 13.
%H Zhi-Wei Sun, <a href="/A219923/b219923.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.
%e a(11)=1 since 11=6+5 with 6-1, 6+1 and 2*6*5+1=61 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[2(Prime[k]+1)(n-Prime[k]-1)+1]==True,1,0],{k,1,PrimePi[n-1]}]
%t Do[Print[n," ",a[n]],{n,1,10000}]
%Y Cf. A219864, A219842.
%K nonn
%O 1,9
%A _Zhi-Wei Sun_, Dec 01 2012
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