A231516 Number of ways to write n = x + y with 0 < x <= y such that x!*y + 1 is prime.

0, 1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 1, 3, 3, 3, 4, 3, 3, 2, 5, 4, 2, 6, 4, 3, 4, 2, 4, 8, 4, 5, 4, 7, 5, 6, 5, 5, 7, 4, 7, 7, 4, 10, 5, 3, 6, 8, 7, 7, 7, 7, 5, 8, 5, 5, 8, 4, 6, 8, 4, 7, 3, 7, 5, 6, 6, 5, 3, 9, 5, 12, 2, 10, 4, 4, 7, 7, 8, 7, 8, 7, 10, 8, 5, 4, 7, 12, 9, 6, 6, 6, 7, 3, 12, 5, 7, 8, 10, 8, 6

Offset: 1

Author

Zhi-Wei Sun, Nov 11 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1. Also, any integer n > 3 can be written as x + y with 0 < x <= y such that x!*y -1 is prime.

We have verified the conjecture for n up to 10^6.

Examples

			a(9) = 1 since 9 = 3 + 6 with 3!*6 + 1 = 37 prime.
a(12) = 1 since 12 = 4 + 8 with 4!*8 + 1 = 193 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000040, A000142, A231201, A231555, A231561, A231557, A231631.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[x!*(n-x)+1],1,0],{x,1,n/2}]
Table[a[n],{n,1,100}]