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A360012 a(n) is the number of triples (u,v,w) of divisors of n with u/v = v/w, and u < v < w.

%I A360012 #8 Jan 22 2023 16:02:36
%S A360012 0,0,0,1,0,0,0,2,1,0,0,2,0,0,0,4,0,2,0,2,0,0,0,4,1,0,2,2,0,0,0,6,0,0,
%T A360012 0,8,0,0,0,4,0,0,0,2,2,0,0,8,1,2,0,2,0,4,0,4,0,0,0,4,0,0,2,9,0,0,0,2,
%U A360012 0,0,0,14,0,0,2,2,0,0,0,8,4,0,0,4,0,0,0
%N A360012 a(n) is the number of triples (u,v,w) of divisors of n with u/v = v/w, and u < v < w.
%C A360012 In other words, a(n) is the number of triples of distinct divisors of n in geometric progression.
%C A360012 This sequence is unbounded.
%H A360012 <a href="/index/Di#divisors">Index entries for sequences related to divisors</a>
%F A360012 a(n) <= a(n*k) for any n, k > 0.
%F A360012 a(p^k) = A002620(k) for any k >= 0 and any prime number p.
%F A360012 a(s^2) = A005059(k) for any squarefree number s with k prime factors.
%e A360012 The first terms, alongside the corresponding triples, are:
%e A360012   n   a(n)  (u,v,w)'s
%e A360012   --  ----  ------------------------------------
%e A360012    1     0  None
%e A360012    2     0  None
%e A360012    3     0  None
%e A360012    4     1  (1,2,4)
%e A360012    5     0  None
%e A360012    6     0  None
%e A360012    7     0  None
%e A360012    8     2  (1,2,4), (2,4,8)
%e A360012    9     1  (1,3,9)
%e A360012   10     0  None
%e A360012   11     0  None
%e A360012   12     2  (1,2,4), (3,6,12)
%e A360012   13     0  None
%e A360012   14     0  None
%e A360012   15     0  None
%e A360012   16     4  (1,2,4), (1,4,16), (2,4,8), (4,8,16)
%t A360012 Array[Count[Subsets[#, {3}], _?(#2 / #1 == #3 / #2 & @@ # &)] &@ Divisors@ # &, 87]
%o A360012 (PARI) a(n) = { my (d=divisors(n), v=0); for (i=1, #d-2, for (j=i+1, #d-1, for (k=j+1, #d, if (d[i]*d[k]==d[j]^2, v++)))); return (v) }
%Y A360012 Cf. A002620, A005059, A091009, A132345.
%K A360012 nonn
%O A360012 1,8
%A A360012 _Rémy Sigrist_, Jan 21 2023