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A368247
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The number of cubefree divisors of the cubefull part of n (A360540).
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2
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1
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OFFSET
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1,8
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LINKS
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FORMULA
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Multiplicative with a(p^e) = 1 if e <= 2, and 3 otherwise.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A073184(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/p^3) = 1.37700168952903630206... .
In general, the asymptotic mean of the number of k-free divisors of the k-full part of n is Product_{p prime} (1 + (k-1)/p^k).
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MATHEMATICA
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f[p_, e_] := If[e > 2, 3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> if(x < 3, 1, 3), factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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