A231635 Number of ways to write n = x + y with 0 < x <= y such that lcm(x, y) + 1 is prime.

0, 1, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 6, 1, 3, 2, 8, 4, 5, 4, 3, 2, 7, 5, 6, 2, 3, 2, 8, 5, 10, 6, 3, 1, 8, 3, 9, 4, 4, 4, 14, 6, 16, 7, 7, 2, 12, 6, 8, 4, 5, 5, 21, 5, 8, 6, 4, 8, 11, 7, 12, 5, 6, 4, 10, 8, 22, 6, 10, 6, 17, 9, 23, 7, 11, 12, 18, 10, 19, 10, 10, 7, 23, 8, 15, 4, 7, 8, 14, 11, 19, 9, 2, 4, 11, 10, 35, 6, 10, 10

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 3 can be written as x + y (x, y > 0) with lcm(x, y) - 1 prime.

(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with lcm(x, y)^2 + lcm(x, y) + 1 prime. Also, any integer n > 1 not equal to 10 can be written as x + y (x, y > 0) with lcm(x, y)^2 + 1 prime.

From Mauro Fiorentini, Aug 02 2023: (Start)

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

Both parts of conjecture (ii) verified for n up to 10^6. (End)

Links

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

Example

a(9) = 1 since 9 = 3 + 6 with lcm(3, 6) + 1 = 7 prime.
a(10) = 1 since 10 = 4 + 6 with lcm(4, 6) + 1 = 13 prime.

Mathematica

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

Crossrefs

Cf. A109909, A219842, A219864, A220272, A231201, A231516, A231555, A231561, A231633.

Sequence in context: A192190 A272729 A064034 * A366281 A365805 A334749

Adjacent sequences: A231632 A231633 A231634 * A231636 A231637 A231638

Keyword

nonn

Author

Zhi-Wei Sun, Nov 12 2013

Status

approved