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

Author

Derek Lim, Apr 08 2020

Keywords

• nonn

Examples

			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).
		

Links

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

Crossrefs

Cf. A000005, A002541, A320224.

Programs

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 *)

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])

Formula

a(n) = Sum_{m=1..n} Sum_{d|m, dAlois P. Heinz, Apr 09 2020