A231555 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).

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn

Comments

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.

(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.

We also have similar conjectures involving some second-order recurrences other than the Fibonacci sequence.

Examples

			a(19) = 1 since 19 = 17 + 2 with 17*18 + F(2) = 307 prime.
a(30) = 1 since 30 = 8 + 22 with 8*9 + F(22) = 17783 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000040, A000045, A231201, A228425, A228428, A228429, A228430, A228431.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[x(x+1)+Fibonacci[n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]