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%I #24 Feb 02 2014 05:59:53
%S 0,0,0,1,0,2,2,2,3,3,1,1,1,3,3,5,2,2,4,1,4,4,4,3,1,3,3,3,7,4,4,2,3,1,
%T 3,5,2,2,5,6,1,6,5,2,4,6,4,3,3,4,4,6,4,3,5,2,3,5,5,2,6,3,6,5,5,4,7,4,
%U 4,8,6,6,5,5,4,8,7,6,4,4,5,6,4,5,4,3,7,9,9,4,8,5,6,2,9,6,7,1,7,8
%N Number of ordered ways to write n = p + q with q > 0 such that p and 2*p + prime(q) are both prime.
%C Conjecture: (i) a(n) > 0 for all n > 5.
%C (ii) Any integer n > 5 can be written as p + q (q > 0) with p and prime(p) + 2*q both prime.
%C (iii) Each integer n > 10 can be written as p + q (q > 0) with p and prime(p)^2 + 4*q^2 both prime.
%H Zhi-Wei Sun, <a href="/A182663/b182663.txt">Table of n, a(n) for n = 1..10000</a>
%e a(11) = 1 since 11 = 5 + 6 with 5 and 2*5 + prime(6) = 10 + 13 = 23 both prime.
%e a(98) = 1 since 98 = 19 + 79 with 19 and 2*19 + prime(79) = 38 + 401 = 439 both prime.
%t p[n_,m_]:=PrimeQ[2*m+Prime[n-m]]
%t a[n_]:=Sum[If[p[n,Prime[k]],1,0],{k,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A182662, A236831.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Feb 01 2014
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