A343877 Number of pairs (d1, d2) of divisors of n such that d1

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

Author

Wesley Ivan Hurt, May 02 2021

Keywords

• nonn

Comments

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

Examples

			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).
		

Crossrefs

Cf. A343824, A000005.

Programs

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}]

PARI

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

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).

a(n) = (1/2)*tau(n)^2 - (3/2)*tau(n) + 1. - Ridouane Oudra, May 29 2025