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%I #6 Nov 11 2013 08:39:25
%S 0,1,1,1,1,2,1,1,2,2,3,3,2,1,4,5,2,2,3,3,2,2,2,4,4,3,5,4,4,3,5,4,5,4,
%T 3,3,2,3,5,5,4,4,3,3,7,5,6,4,6,5,4,6,5,5,5,3,5,6,7,8,4,4,3,4,2,3,5,6,
%U 7,7,4,3,6,6,6,8,3,4,7,7,6,6,5,7,6,7,8,5,6,5,7,2,5,5,7,5,7,6,10,8
%N Number of ways to write n = x + y with 0 < x <= y such that 2^x * y + 1 is prime.
%C Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as x + y (x, y > 0) with 2^x * y^2 + 1 prime.
%C (ii) Each integer n > 2 can be written as x + y (x, y > 0) with 2^x * y - 1 prime. Also, every n = 3, 4, ... can be expressed as x + y (x, y > 0) with 2^x * y^2 - 1 prime.
%H Zhi-Wei Sun, <a href="/A231561/b231561.txt">Table of n, a(n) for n = 1..4000</a>
%e a(7) = 1 since 7 = 1 + 6 with 2^1 * 6 + 1 = 13 prime.
%e a(14) = 1 since 14 = 3 + 11 with 2^3 * 11 + 1 = 89 prime.
%t a[n_]:=Sum[If[PrimeQ[2^x*(n-x)+1],1,0],{x,1,n/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000079, A228425, A231201, A231555, A231557.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Nov 11 2013
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