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%I #5 Nov 11 2013 08:37:27
%S 0,1,2,1,2,2,2,2,4,3,2,2,3,3,3,3,6,3,4,2,5,3,1,4,4,3,4,3,2,4,6,3,3,7,
%T 4,7,6,5,4,5,3,7,3,4,6,6,3,4,7,4,8,6,5,11,5,5,9,7,4,7,8,5,3,1,6,5,8,4,
%U 7,5,2,8,8,7,4,3,8,7,3,3,8,8,4,8,8,5,5,7,8,6,7,8,11,6,7,9,7,6,2,3
%N Number of ways to write n = x + y (x, y > 0) with 2^x + y*(y+1)/2 prime.
%C Conjecture: a(n) > 0 for all n > 1.
%C This implies that there are infinitely many primes each of which is a sum of a power of 2 and a triangular number.
%C See also A231201, A231555 and A231561 for other similar conjectures.
%H Zhi-Wei Sun, <a href="/A231577/b231577.txt">Table of n, a(n) for n = 1..7000</a>
%e a(23) = 1 since 23 = 9 + 14 with 2^9 + 14*15/2 = 617 prime.
%e a(64) = 1 since 64 = 14 + 50 with 2^{14} + 50*51/2 = 17659 prime.
%t a[n_]:=Sum[If[PrimeQ[2^x+(n-x)(n-x+1)/2],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000079, A000217, A231201, A231555, A231557, A231561.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Nov 11 2013