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

%I A333885 #27 Nov 27 2020 07:41:33
%S A333885 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,
%T A333885 65,72,72,84,84,94,96,98,100,119,119,121,123,138,138,150,150,157,164,
%U A333885 166,166,192,193,200,202,209,209,224,226,241,243,245,245,276,276
%N A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.
%H A333885 Alois P. Heinz, <a href="/A333885/b333885.txt">Table of n, a(n) for n = 1..10000</a>
%F A333885 a(n) = Sum_{m=1..n} Sum_{d|m, d<m} (tau(d)-1). - _Alois P. Heinz_, Apr 09 2020
%e A333885 The a(4) = 1 triple is (1,2,4).
%e A333885 The a(8) = 6 triples are (1,2,4), (1,2,6), (1,2,8), (1,3,6), (1,4,8), (2,4,8).
%p A333885 with(numtheory):
%p A333885 a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
%p A333885        add(tau(d)-1, d=divisors(n) minus {n}))
%p A333885     end:
%p A333885 seq(a(n), n=1..80);  # _Alois P. Heinz_, Apr 09 2020
%t A333885 a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sum[DivisorSigma[0, d] - 1, {d, Most @ Divisors[n]}]];
%t A333885 Array[a, 80] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o A333885 (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])
%Y A333885 Cf. A000005, A002541, A320224.
%K A333885 nonn
%O A333885 1,6
%A A333885 _Derek Lim_, Apr 08 2020