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A369719 The number of divisors of the smallest cubefull number that is a multiple of n.

%I A369719 #13 Jan 30 2024 05:20:05
%S A369719 1,4,4,4,4,16,4,4,4,16,4,16,4,16,16,5,4,16,4,16,16,16,4,16,4,16,4,16,
%T A369719 4,64,4,6,16,16,16,16,4,16,16,16,4,64,4,16,16,16,4,20,4,16,16,16,4,16,
%U A369719 16,16,16,16,4,64,4,16,16,7,16,64,4,16,16,64,4,16,4
%N A369719 The number of divisors of the smallest cubefull number that is a multiple of n.
%H A369719 Amiram Eldar, <a href="/A369719/b369719.txt">Table of n, a(n) for n = 1..10000</a>
%F A369719 a(n) = A000005(A356193(n)).
%F A369719 Multiplicative with a(p) = 4 for e <= 2, and a(p^e) = e+1 for e >= 3.
%F A369719 a(n) >= A000005(n), with equality if and only if n is cubefull (A036966).
%F A369719 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^s - 3/p^(2*s) + 1/p^(4*s)).
%F A369719 Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + 1/p^(6*s) - 2/p^(5*s) - 2/p^(4*s) + 8/p^(3*s) - 6/p^(2*s)). - _Vaclav Kotesovec_, Jan 30 2024
%t A369719 f[p_, e_] := If[e <= 2, 4, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369719 (PARI) a(n) = vecprod(apply(x -> if(x <= 2, 4, x+1), factor(n)[, 2]));
%o A369719 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 * ((1 + 2*X - 3*X^2 + X^4)))[n], ", ")) \\ _Vaclav Kotesovec_, Jan 30 2024
%Y A369719 Cf. A000005, A036966, A356193, A369716, A369720, A369721.
%K A369719 nonn,easy,mult
%O A369719 1,2
%A A369719 _Amiram Eldar_, Jan 30 2024