A360012 - OEIS

%I #8 Jan 22 2023 16:02:36

%S 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 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 0,0,0,14,0,0,2,2,0,0,0,8,4,0,0,4,0,0,0

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

%C In other words, a(n) is the number of triples of distinct divisors of n in geometric progression.

%C This sequence is unbounded.

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors</a>

%F a(n) <= a(n*k) for any n, k > 0.

%F a(p^k) = A002620(k) for any k >= 0 and any prime number p.

%F a(s^2) = A005059(k) for any squarefree number s with k prime factors.

%e The first terms, alongside the corresponding triples, are:

%e   n   a(n)  (u,v,w)'s

%e   --  ----  ------------------------------------

%e    1     0  None

%e    2     0  None

%e    3     0  None

%e    4     1  (1,2,4)

%e    5     0  None

%e    6     0  None

%e    7     0  None

%e    8     2  (1,2,4), (2,4,8)

%e    9     1  (1,3,9)

%e   10     0  None

%e   11     0  None

%e   12     2  (1,2,4), (3,6,12)

%e   13     0  None

%e   14     0  None

%e   15     0  None

%e   16     4  (1,2,4), (1,4,16), (2,4,8), (4,8,16)

%t Array[Count[Subsets[#, {3}], _?(#2 / #1 == #3 / #2 & @@ # &)] &@ Divisors@ # &, 87]

%o (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 Cf. A002620, A005059, A091009, A132345.

%K nonn

%O 1,8

%A _Rémy Sigrist_, Jan 21 2023