A343879 Number of pairs (d1, d2) of divisors of n such that d1<d2, d1|n, d2|n, d1|d2 and d1 + d2 <= n.

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 7, 0, 2, 2, 6, 0, 7, 0, 7, 2, 2, 0, 15, 1, 2, 3, 7, 0, 12, 0, 10, 2, 2, 2, 19, 0, 2, 2, 15, 0, 12, 0, 7, 7, 2, 0, 26, 1, 7, 2, 7, 0, 15, 2, 15, 2, 2, 0, 31, 0, 2, 7, 15, 2, 12, 0, 7, 2, 12, 0, 37, 0, 2, 7, 7, 2, 12, 0, 26, 6, 2, 0, 31, 2, 2, 2, 15

Offset

1,6

Comments

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

Links

Table of n, a(n) for n=1..88.

Formula

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

Example

a(12) = 7; The 7 pairs are (1,2), (1,3), (1,4), (1,6), (2,4), (2,6), (3,6).

Mathematica

Table[Sum[Sum[(1 - Ceiling[k/i] + Floor[k/i]) (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) && !(d2 % d1), 1))); \\ Michel Marcus, May 02 2021

Crossrefs

Cf. A343877.

Sequence in context: A071465 A333409 A197117 * A275387 A051709 A318326

Adjacent sequences: A343876 A343877 A343878 * A343880 A343881 A343882

Keyword

nonn

Author

Wesley Ivan Hurt, May 02 2021

Status

approved