cp's OEIS Frontend

A368248 The number of unitary divisors of the cubefull part of n (A360540).

%I A368248 #19 Jun 19 2025 05:56:02
%S A368248 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T A368248 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A368248 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A368248 The number of unitary divisors of the cubefull part of n (A360540).
%C A368248 First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.
%C A368248 Also, the number of squarefree divisors of the cubefull part of n.
%C A368248 Also, the number of cubes of squarefree numbers (A062838) that divide n.
%C A368248 The number of unitary divisors of n that are cubefull numbers (A036966). - _Amiram Eldar_, Jun 19 2025
%H A368248 Amiram Eldar, <a href="/A368248/b368248.txt">Table of n, a(n) for n = 1..10000</a>
%F A368248 a(n) = A034444(A360540(n)).
%F A368248 a(n) = abs(A307428(n)).
%F A368248 Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.
%F A368248 a(n) >= 1, with equality if and only if n is cubefree (A004709).
%F A368248 a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).
%F A368248 Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).
%F A368248 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).
%F A368248 In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).
%t A368248 f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368248 (PARI) a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));
%Y A368248 Cf. A004709, A034444, A036966, A062838, A157289, A307428, A323308, A360540, A366145.
%Y A368248 Cf. A061704, A304327, A362852.
%K A368248 nonn,easy,mult
%O A368248 1,8
%A A368248 _Amiram Eldar_, Dec 19 2023