A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.

0, 0, 0, 1, 1, 3, 3, 6, 7, 9, 9, 16, 16, 18, 20, 26, 26, 33, 33, 40, 42, 44, 44, 59, 60, 62, 65, 72, 72, 84, 84, 94, 96, 98, 100, 119, 119, 121, 123, 138, 138, 150, 150, 157, 164, 166, 166, 192, 193, 200, 202, 209, 209, 224, 226, 241, 243, 245, 245, 276, 276

Offset

1,6

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{m=1..n} Sum_{d|m, d<m} (tau(d)-1). - Alois P. Heinz, Apr 09 2020

Example

The a(4) = 1 triple is (1,2,4).
The a(8) = 6 triples are (1,2,4), (1,2,6), (1,2,8), (1,3,6), (1,4,8), (2,4,8).

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
       add(tau(d)-1, d=divisors(n) minus {n}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 09 2020

Mathematica

a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sum[DivisorSigma[0, d] - 1, {d, Most @ Divisors[n]}]];
Array[a, 80] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

Prog

(Python) an = len([(i, j, k) for i in range(1, n+1) for j in range(i+1, n+1) for k in range(j+1, n+1) if j%i==0 and k%j==0])

Crossrefs

Cf. A000005, A002541, A320224.

Sequence in context: A273062 A269170 A003879 * A270060 A350636 A078565

Adjacent sequences: A333882 A333883 A333884 * A333886 A333887 A333888

Keyword

nonn

Author

Derek Lim, Apr 08 2020

Status

approved