%I #8 Jan 28 2014 09:24:01
%S 1,1,0,1,1,2,0,1,1,0,3,2,1,2,2,1,1,2,1,1,0,2,2,4,0,4,2,1,2,3,0,4,3,2,
%T 1,1,1,1,2,2,1,3,2,2,5,1,5,3,3,4,5,1,4,1,3,3,6,4,4,1,4,4,3,5,5,5,2,2,
%U 2,4,3,2,3,3,7,4,3,2,4,3,5,3,3,5,2,4,6,3,4,3,4,2,5,2,7,6,3,3,5,4
%N Number of ways to write 2*n = k + m with 0 < k <= m such that prime(k) + m and k + prime(m) are both prime.
%C Conjecture: (i) a(n) > 0 for all n > 31.
%C (ii) Any positive even number can be written as k + m with k > 0 and m > 0 such that (prime(k) - k) + m and k + (prime(m) - m) are both prime.
%C (iii) Each integer n > 24 can be written as k + m with k > 0 and m > 0 such that prime(k) + m*(m-1) and k*(k-1) + prime(m) are both prime.
%C (iv) Any integer n > 15 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m) and phi(k) + prime(m) are both prime.
%C Part (ii) of the conjecture in A232443 implies that any integer n > 7 can be written as k + m (k > 0, m > 0) with prime(k) + m = n + prime(k) - k prime.
%H Zhi-Wei Sun, <a href="/A236541/b236541.txt">Table of n, a(n) for n = 1..10000</a>
%e a(9) = 1 since 2*9 = 8 + 10 with prime(8) + 10 = 19 + 10 = 29 and 8 + prime(10) = 8 + 29 = 37 both prime.
%e a(92) = 1 since 2*92 = 86 + 98 with prime(86) + 98 = 443 + 98 = 541 and 86 + prime(98) = 86 + 521 = 607 both prime.
%t p[k_,m_]:=PrimeQ[Prime[k]+m]
%t a[n_]:=Sum[If[p[k,2n-k]&&p[2n-k,k],1,0],{k,1,n}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A232443, A232463.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Jan 28 2014