cp's OEIS Frontend

A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

%I A369306 #12 Apr 26 2025 20:54:07
%S A369306 1,2,2,3,2,4,2,2,3,4,2,6,2,4,4,1,2,6,2,6,4,4,2,4,3,4,2,6,2,8,2,0,4,4,
%T A369306 4,9,2,4,4,4,2,8,2,6,6,4,2,2,3,6,4,6,2,4,4,4,4,4,2,12,2,4,6,0,4,8,2,6,
%U A369306 4,8,2,6,2,4,6,6,4,8,2,2,1,4,2,12,4,4,4
%N A369306 The number of cubefree divisors d of n such that n/d is also cubefree.
%C A369306 The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)).
%H A369306 Amiram Eldar, <a href="/A369306/b369306.txt">Table of n, a(n) for n = 1..10000</a>
%F A369306 Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5.
%F A369306 a(n) >= 0, with equality if and only if n is a 5-full number (A069492) larger than 1.
%F A369306 a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117).
%F A369306 a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).
%F A369306 Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2.
%F A369306 Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma - 1 - 6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).
%t A369306 f[p_,e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369306 (PARI) a(n) = vecprod(apply(x -> [2, 3, 2, 1, 0][min(x, 5)], factor(n)[,2]));
%Y A369306 Cf. A000005, A004709, A005117, A007427, A069492, A073184.
%Y A369306 Cf. A001620, A002117, A244115.
%K A369306 nonn,easy,mult
%O A369306 1,2
%A A369306 _Amiram Eldar_, Jan 19 2024