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A235330 Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.

%I A235330 #8 Jan 06 2014 11:14:41
%S A235330 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 A235330 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 A235330 2,3,3,3,1,1,3,3,2,3,1,1,5,0,1,4,2,1,1
%N A235330 Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.
%C A235330 Conjecture: (i) a(n) > 0 for all n >= 2480.
%C A235330 (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 A235330 Parts (i) and (ii) are stronger than Goldbach's conjecture  (A045917) and Lemoine's conjecture (A046927) respectively.
%H A235330 Zhi-Wei Sun, <a href="/A235330/b235330.txt">Table of n, a(n) for n = 1..10000</a>
%e A235330 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 A235330 p[n_] := PrimeQ[n] && PrimeQ[Prime[n] - n + 1];
%t A235330 q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];
%t A235330 a[n_] := Sum[If[p[k] && q[2 n - k], 1, 0], {k, 1, 2 n - 1}];
%t A235330 Table[a[n], {n, 1, 100}]
%Y A235330 Cf. A000040, A045917, A046927, A234695, A235187, A235189.
%K A235330 nonn
%O A235330 1,9
%A A235330 _Zhi-Wei Sun_, Jan 05 2014