A343877 Number of pairs (d1, d2) of divisors of n such that d1<d2, d1|n, d2|n, and d1 + d2 <= n.

0, 0, 0, 1, 0, 3, 0, 3, 1, 3, 0, 10, 0, 3, 3, 6, 0, 10, 0, 10, 3, 3, 0, 21, 1, 3, 3, 10, 0, 21, 0, 10, 3, 3, 3, 28, 0, 3, 3, 21, 0, 21, 0, 10, 10, 3, 0, 36, 1, 10, 3, 10, 0, 21, 3, 21, 3, 3, 0, 55, 0, 3, 10, 15, 3, 21, 0, 10, 3, 21, 0, 55, 0, 3, 10, 10, 3, 21, 0, 36, 6, 3, 0, 55

Offset

1,6

Comments

a(n) = 0 if and only if n is noncomposite.

Links

Table of n, a(n) for n=1..84.

Formula

a(n) = Sum_{k=1..floor(n/2)} Sum_{i=1..k-1} c(n/k) * c(n/i), where c(n) = 1 - ceiling(n) + floor(n).

Example

a(12) = 10; The 10 pairs are (1,2), (1,3), (1,4), (1,6), (2,3), (2,4), (2,6), (3,4), (3,6), (4,6).

Mathematica

Table[Sum[Sum[(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sumdiv(n, d1, sumdiv(n, d2, if ((d1 < d2) && (d1+d2 <= n), 1))); \\ Michel Marcus, May 02 2021

Crossrefs

Cf. A343824.

Sequence in context: A307807 A350849 A318504 * A318505 A376627 A237595

Adjacent sequences: A343874 A343875 A343876 * A343878 A343879 A343880

Keyword

nonn

Author

Wesley Ivan Hurt, May 02 2021

Status

approved