|
%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
|
