A231577 Number of ways to write n = x + y (x, y > 0) with 2^x + y*(y+1)/2 prime.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 11 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

This implies that there are infinitely many primes each of which is a sum of a power of 2 and a triangular number.

See also A231201, A231555 and A231561 for other similar conjectures.

Examples

			 a(23) = 1 since 23 = 9 + 14 with 2^9 + 14*15/2 = 617 prime.
a(64) = 1 since 64 = 14 + 50 with 2^{14} + 50*51/2 = 17659 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Crossrefs

Cf. A000040, A000079, A000217, A231201, A231555, A231557, A231561.

Programs

Mathematica

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