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

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n.

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)

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A007412, A056595, A061704.

Cf. A001620, A002117.

Sequence in context: A323014 A350331 A293227 * A241165 A233183 A035942

Adjacent sequences: A291205 A291206 A291207 * A291209 A291210 A291211

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 21 2017

Status

approved