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A365487 The number of divisors of the largest cube dividing n.

%I A365487 #8 Sep 06 2023 01:22:28
%S A365487 1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,4,1,1,1,1,4,1,1,
%T A365487 1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,1,1,1,4,1,4,1,1,1,1,1,1,1,7,1,1,1,1,
%U A365487 1,1,1,4,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1
%N A365487 The number of divisors of the largest cube dividing n.
%C A365487 The number of divisors of the cube root of the largest cube dividing n, A053150(n), is A061704(n).
%H A365487 Amiram Eldar, <a href="/A365487/b365487.txt">Table of n, a(n) for n = 1..10000</a>
%F A365487 a(n) = A000005(A008834(n)).
%F A365487 Multiplicative with a(p^e) = 3*floor(e/3) + 1.
%F A365487 a(n) = 1 if and only if n is cubefree (A004709).
%F A365487 a(n) <= A000005(n) with equality if and only if n is a cube (A000578).
%F A365487 Dirichlet g.f.: zeta(s) * zeta(3*s) * Product_{p prime} (1 + 2/p^(3*s)).
%F A365487 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 2/p^3) = 1.6552343865608... .
%t A365487 f[p_, e_] := 3*Floor[e/3] + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
%o A365487 (PARI) a(n) = vecprod(apply(x -> 3*(x\3) + 1, factor(n)[, 2]));
%Y A365487 Cf. A000005, A000578, A004709, A008834, A053150, A061704.
%K A365487 nonn,easy,mult
%O A365487 1,8
%A A365487 _Amiram Eldar_, Sep 05 2023