A291208 Number of noncube divisors of n.

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 6, 2, 3, 2, 5, 1, 7, 1, 4, 3, 3, 3, 8, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 8, 2, 5, 3, 5, 1, 6, 3, 6, 3, 3, 1, 11, 1, 3, 5, 4, 3, 7, 1, 5, 3, 7, 1, 10, 1, 3, 5, 5, 3, 7, 1, 8, 3, 3, 1, 11, 3, 3, 3, 6, 1, 11, 3, 5, 3, 3, 3, 10, 1, 5, 5, 8, 1, 7, 1, 6, 7

Offset: 1

Author

Ilya Gutkovskiy, Aug 21 2017

Keywords

• nonn

Examples

			a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are noncubes {2, 4}.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A007412, A056595, A061704.

Cf. A001620, A002117.

Programs

Mathematica

nmax = 105; Rest[CoefficientList[Series[Sum[(x^k - x^k^3)/((1 - x^k) (1 - x^k^3)), {k, 1, nmax}], {x, 0, nmax}], x]]
f1[p_, e_] := e + 1; f2[p_, e_] := 1 + Floor[e/3]; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f1 @@@ fct - Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

PARI

a(n) = sumdiv(n, d, !ispower(d, 3)); \\ Michel Marcus, Aug 21 2017

Python

from math import prod
from sympy import factorint
def A291208(n):
    f = factorint(n).values()
    return prod(e+1 for e in f)-prod(e//3+1 for e in f) # Chai Wah Wu, Jun 05 2025

Formula

G.f.: Sum_{k>=1} x^A007412(k)/(1 - x^A007412(k)).

G.f.: Sum_{k>=1} (x^k - x^(k^3))/((1 - x^k)*(1 - x^(k^3))).

a(n) = A000005(n) - A061704(n).

From Amiram Eldar, Jan 30 2025: (Start)

Dirichlet g.f.: zeta(s) * (zeta(s) - zeta(3*s)).

Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(3) - 1), where gamma is Euler's constant (A001620). (End)