%I #5 Feb 26 2014 07:50:25
%S 0,0,1,0,1,2,1,0,0,0,1,1,2,2,1,2,1,2,3,2,3,3,2,3,2,2,2,2,3,1,1,2,2,3,
%T 2,1,2,1,1,3,3,2,1,1,3,3,5,5,2,2,2,3,4,5,5,4,3,2,2,2,3,2,3,4,1,3,4,3,
%U 4,6,7,6,3,2,2,2,3,4,5,3
%N a(n) = |{0 < k < n-1: p = prime(k) + pi(n-k) and p + 2 are both prime}|, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 10.
%C (ii) For any integer n > 4, there is a positive integer k < n such that prime(k)^2 + pi(n-k)^2 is prime.
%C We have verified part (i) of the conjecture for n up to 10^7.
%H Zhi-Wei Sun, <a href="/A238386/b238386.txt">Table of n, a(n) for n = 1..10000</a>
%e a(7) = 1 since prime(1) + pi(7-1) = 2 + 3 = 5 and 5 + 2 = 7 are both prime.
%e a(30) = 1 since prime(16) + pi(30-16) = 53 + 6 = 59 and 59 + 2 are both prime.
%e a(108) = 1 since prime(15) + pi(108-15) = 47 + 24 = 71 and 71 + 2 = 73 are both prime.
%t tq[n_]:=PrimeQ[n]&&PrimeQ[n+2]
%t a[n_]:=Sum[If[tq[Prime[k]+PrimePi[n-k]],1,0],{k,1,n-2}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000720, A001359, A006512.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Feb 26 2014