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A360012
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a(n) is the number of triples (u,v,w) of divisors of n with u/v = v/w, and u < v < w.
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0
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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
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OFFSET
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1,8
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COMMENTS
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In other words, a(n) is the number of triples of distinct divisors of n in geometric progression.
This sequence is unbounded.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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)
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MATHEMATICA
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Array[Count[Subsets[#, {3}], _?(#2 / #1 == #3 / #2 & @@ # &)] &@ Divisors@ # &, 87]
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PROG
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(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) }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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