A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.

0, 0, 1, 2, 3, 1, 3, 2, 5, 2, 4, 2, 7, 2, 5, 3, 5, 3, 10, 4, 5, 3, 8, 3, 14, 6, 5, 4, 11, 5, 11, 3, 11, 9, 4, 5, 10, 5, 11, 9, 12, 3, 19, 7, 11, 6, 12, 9, 11, 7, 17, 7, 13, 5, 22, 3, 3, 15, 16, 5, 25, 4, 9, 11, 13, 5, 19, 6, 22, 6, 11, 6, 39, 6, 24, 7, 7, 6, 25, 8, 21, 11, 24, 7, 31, 7, 19, 11, 33, 10, 14, 8, 15, 27, 18, 9, 21, 4, 27, 9

Offset

1,4

Comments

Conjectures:

(i) a(n) > 0 for all n > 2. Also, any integer n > 4 can be written as x + y (x, y > 0) with x^2 * y + 1 prime.

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

(iii) Also, any integer n > 2 can be written as x + y (x, y > 0) with 2*(x*y)^2 - 1 (or (x*y)^2 + x*y - 1) prime.

From Mauro Fiorentini, Jul 31 2023: (Start)

Both parts of conjecture (i) verified for n up to 10^9.

Conjecture (ii) and both parts of conjecture (iii) verified for n up to 10^7. (End)

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(6) = 1 since 6 = 4 + 2 with 4^2*2 - 1 = 31 prime.

Mathematica

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

Crossrefs

Cf. A000040, A000290, A109909, A219842, A219864, A220272, A228424, A231201, A231155, A231561, A231557, A231631.

Sequence in context: A194308 A257070 A289171 * A039702 A230294 A104483

Adjacent sequences: A231630 A231631 A231632 * A231634 A231635 A231636

Keyword

nonn

Author

Zhi-Wei Sun, Nov 12 2013

Status

approved