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A368247 The number of cubefree divisors of the cubefull part of n (A360540).

%I A368247 #10 Dec 19 2023 09:19:36
%S A368247 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,
%T A368247 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,
%U A368247 1,1,1,3,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1
%N A368247 The number of cubefree divisors of the cubefull part of n (A360540).
%H A368247 Amiram Eldar, <a href="/A368247/b368247.txt">Table of n, a(n) for n = 1..10000</a>
%F A368247 a(n) = A073184(A360540(n)).
%F A368247 Multiplicative with a(p^e) = 1 if e <= 2, and 3 otherwise.
%F A368247 a(n) >= 1, with equality if and only if n is cubefree (A004709).
%F A368247 a(n) <= A073184(n), with equality if and only if n is cubefull (A036966).
%F A368247 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/p^(3*s)).
%F A368247 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/p^3) = 1.37700168952903630206... .
%F A368247 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).
%t A368247 f[p_, e_] := If[e > 2, 3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368247 (PARI) a(n) = vecprod(apply(x -> if(x < 3, 1, 3), factor(n)[, 2]));
%Y A368247 Cf. A004709, A036966, A073184, A323308, A360540.
%K A368247 nonn,easy,mult
%O A368247 1,8
%A A368247 _Amiram Eldar_, Dec 19 2023