A231561 Number of ways to write n = x + y with 0 < x <= y such that 2^x * y + 1 is prime.

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

Offset

1,6

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 2^x * y^2 + 1 prime.

(ii) Each integer n > 2 can be written as x + y (x, y > 0) with 2^x * y - 1 prime. Also, every n = 3, 4, ... can be expressed as x + y (x, y > 0) with 2^x * y^2 - 1 prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Example

a(7) = 1 since 7 = 1 + 6 with 2^1 * 6 + 1 = 13 prime.
a(14) = 1 since 14 = 3 + 11 with 2^3 * 11 + 1 = 89 prime.

Mathematica

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

Crossrefs

Cf. A000040, A000079, A228425, A231201, A231555, A231557.

Sequence in context: A351192 A306460 A317805 * A371632 A113297 A119985

Adjacent sequences: A231558 A231559 A231560 * A231562 A231563 A231564

Keyword

nonn

Author

Zhi-Wei Sun, Nov 11 2013

Status

approved