%I
%S 0,0,0,1,1,1,1,1,2,0,1,1,1,3,1,1,2,3,0,1,2,0,3,1,0,2,2,0,0,1,1,2,3,0,
%T 1,3,0,2,0,0,2,1,0,1,2,0,3,0,0,4,2,1,1,1,1,3,4,1,1,3,1,0,2,1,1,3,0,0,
%U 2,3,3,3,1,1,3,3,2,3,1,1,5,0,1,4,2,1,1
%N Number of ways to write 2*n = p + q with p, q, prime(p)  p + 1 and prime(q) + q + 1 all prime.
%C Conjecture: (i) a(n) > 0 for all n >= 2480.
%C (ii) If n > 4368 then 2*n+1 can be written as 2*p + q with p and q terms of the sequence A234695.
%C Parts (i) and (ii) are stronger than Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927) respectively.
%H ZhiWei Sun, <a href="/A235330/b235330.txt">Table of n, a(n) for n = 1..10000</a>
%e a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5)  5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.
%t p[n_] := PrimeQ[n] && PrimeQ[Prime[n]  n + 1];
%t q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];
%t a[n_] := Sum[If[p[k] && q[2 n  k], 1, 0], {k, 1, 2 n  1}];
%t Table[a[n], {n, 1, 100}]
%Y Cf. A000040, A045917, A046927, A234695, A235187, A235189.
%K nonn
%O 1,9
%A _ZhiWei Sun_, Jan 05 2014
