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

1,3

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A168557 A328204 A194320 * A120425 A104186 A350399

Adjacent sequences: A231552 A231553 A231554 * A231556 A231557 A231558

Keyword

nonn

Author

Zhi-Wei Sun, Nov 10 2013

Status

approved