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A380395 The number of unitary divisors of n that are cubes.

%I A380395 #11 Jan 25 2025 09:14:35
%S A380395 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,
%T A380395 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A380395 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A380395 The number of unitary divisors of n that are cubes.
%C A380395 First differs from A318672 at n = 64.
%C A380395 The sum of unitary divisors of n that are cubes is A380396(n).
%H A380395 Amiram Eldar, <a href="/A380395/b380395.txt">Table of n, a(n) for n = 1..10000</a>
%F A380395 a(n) = Sum_{d|n, gcd(d, n/d) = 1} [d is cube], where [] is the Iverson bracket.
%F A380395 Multiplicative with a(p^e) = 2 is e is divisible by 3, and 1 otherwise.
%F A380395 a(n) = abs(A307427(n)).
%F A380395 a(n) = A061704(n) - A380397(n).
%F A380395 a(n) >= 1, with equality if and only if n is not in A366761.
%F A380395 a(n) <= A061704(n), with equality if and only if n is biquadratefree (A046100).
%F A380395 Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(4*s).
%F A380395 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(4) = 1.11062653532614811717... .
%F A380395 In general, the asymptotic mean of the number of unitary divisors of n that are m-th powers is zeta(m)/zeta(m+1), for m >= 2.
%e A380395 a(8) = 2 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3.
%e A380395 a(216) = 4 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3.
%t A380395 f[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380395 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1));}
%Y A380395 Cf. A000578, A046100, A056624, A061704, A077610, A307427, A318672, A366761, A380396, A380397.
%Y A380395 Cf. A002117, A013662.
%K A380395 nonn,easy,mult
%O A380395 1,8
%A A380395 _Amiram Eldar_, Jan 23 2025