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

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, 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, 0, 0, 0, 14, 0, 0, 2, 2, 0, 0, 0, 8, 4, 0, 0, 4, 0, 0, 0

Offset: 1

Author

Rémy Sigrist, Jan 21 2023

Keywords

• nonn

Comments

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

This sequence is unbounded.

Examples

			The first terms, alongside the corresponding triples, are:
  n   a(n)  (u,v,w)'s
  --  ----  ------------------------------------
   1     0  None
   2     0  None
   3     0  None
   4     1  (1,2,4)
   5     0  None
   6     0  None
   7     0  None
   8     2  (1,2,4), (2,4,8)
   9     1  (1,3,9)
  10     0  None
  11     0  None
  12     2  (1,2,4), (3,6,12)
  13     0  None
  14     0  None
  15     0  None
  16     4  (1,2,4), (1,4,16), (2,4,8), (4,8,16)
		

Links

• Index entries for sequences related to divisors

Crossrefs

Cf. A002620, A005059, A091009, A132345.

Programs

Mathematica

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

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) }

Formula

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

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

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