A380397 The number of nonunitary divisors of n that are cubes.

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, 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, 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

Offset

1

Comments

First differs from A295884 at n = 128.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

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

Example

a(16) = 1 since 16 has 1 nonunitary divisor that is a cube, 8 = 2^3.
a(128) = 2 since 128 has 2 nonunitary divisors that are cubes, 8 = 2^3 and 64 = 4^3.

Mathematica

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]

Prog

(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)); }

Crossrefs

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

Cf. A002117, A013662.

Sequence in context: A277164 A011730 A358261 * A295884 A380454 A101637

Adjacent sequences: A380394 A380395 A380396 * A380398 A380399 A380400

Keyword

nonn,easy

Author

Amiram Eldar, Jan 23 2025

Status

approved