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%I #14 Nov 11 2013 03:22:38
%S 0,1,2,2,2,2,3,3,3,1,2,4,2,3,4,6,3,5,1,3,5,6,6,4,5,5,4,7,5,1,5,6,6,6,
%T 6,6,8,6,5,5,5,5,6,3,4,8,9,8,4,5,8,8,6,5,9,5,9,8,8,6,9,7,8,7,6,4,8,7,
%U 8,11,6,7,9,4,5,8,8,7,10,10,11,9,3,5,6,6,4,12,5,9,12,11,7,6,7,9,6,10,5,6
%N Number of ways to write n = x + y (x, y > 0) with x*(x + 1) + F(y) prime, where F(y) denotes the y-th Fibonacci number (A000045).
%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 x + F(y) prime.
%C (ii) Each positive integer n not among 1, 7, 55 can be written as x + y (x, y > 0) with x*(x+1)/2 + F(y) prime. Also, any positive integer n not among 1, 10, 13, 20, 255 can be written as x + y (x, y > 0) with x^2 + F(y) prime.
%C We also have similar conjectures involving some second-order recurrences other than the Fibonacci sequence.
%H Zhi-Wei Sun, <a href="/A231555/b231555.txt">Table of n, a(n) for n = 1..5000</a>
%e a(19) = 1 since 19 = 17 + 2 with 17*18 + F(2) = 307 prime.
%e a(30) = 1 since 30 = 8 + 22 with 8*9 + F(22) = 17783 prime.
%t a[n_]:=Sum[If[PrimeQ[x(x+1)+Fibonacci[n-x]],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000045, A231201, A228425, A228428, A228429, A228430, A228431.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Nov 10 2013
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