A380397 - OEIS

%I #10 Jan 25 2025 08:29:36

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0

%N The number of nonunitary divisors of n that are cubes.

%C First differs from A295884 at n = 128.

%H Amiram Eldar, <a href="/A380397/b380397.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, gcd(d, n/d) > 1} [d is cube], where [] is the Iverson bracket.

%F a(n) = A061704(n) - A380395(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)*(1 - 1/zeta(4)) = 0.091430367833446168224... .

%e a(16) = 1 since 16 has 1 nonunitary divisor that is a cube, 8 = 2^3.

%e a(128) = 2 since 128 has 2 nonunitary divisors that are cubes, 8 = 2^3 and 64 = 4^3.

%t f1[p_, e_] := 1 + Floor[e/3]; f2[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n])- Times @@ f2 @@@ fct; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 2]\3) - prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1));}

%Y Cf. A000578, A048105, A056626, A061704, A295884, A380395, A380396.

%Y Cf. A002117, A013662.

%K nonn,easy

%O 1

%A _Amiram Eldar_, Jan 23 2025