A113061 Sum of cube divisors of n.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 73, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9

Offset

1,8

Comments

Multiplicative with a(p^e) = (p^(3*(1+floor(e/3)))-1)/(p^3-1). The Dirichlet generating function is zeta(s)*zeta(3s-3). The sequence is the inverse Mobius transform of n*A010057(n). - R. J. Mathar, Feb 18 2011

Links

Antti Karttunen, Table of n, a(n) for n = 1..19683

R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.

Index entries for sequences related to sums of divisors

Formula

G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)). - Ilya Gutkovskiy, Dec 24 2016

a(n) = Sum_{d|n} A010057(d)*d. - Antti Karttunen, Oct 08 2017

Sum_{k=1..n} a(k) ~ zeta(4/3)*n^(4/3)/4 - n/2. - Vaclav Kotesovec, Dec 01 2020

Maple

A113061 := proc(n)
    local a, pe, p, e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        e := 3*(1+floor(e/3)) ;
        a := a*(p^e-1)/(p^3-1) ;
    end do:
    a ;
end proc:
seq(A113061(n), n=1..100) ; # R. J. Mathar, Oct 08 2017

Mathematica

a[n_] := Sum[If[IntegerQ[d^(1/3)], d, 0], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 25 2017 *)
f[p_, e_] := (p^(3*(1 + Floor[e/3])) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

Prog

(PARI) A113061(n) = sumdiv(n, d, ispower(d, 3)*d); \\ Antti Karttunen, Oct 08 2017
(Scheme)
;; With memoization-macro definec, after the multiplicative formula of R. J. Mathar:
(definec (A113061 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (+ -1 (expt p (* 3 (+ 1 (A002264 e))))) (+ -1 (expt p 3))) (A113061 (A028234 n)))))) ;; Antti Karttunen, Oct 08 2017

Crossrefs

Cf. A004709, A010057, A035316.

Sequence in context: A189788 A264981 A370240 * A366904 A284099 A176410

Adjacent sequences: A113058 A113059 A113060 * A113062 A113063 A113064

Keyword

nonn,mult

Author

Paul Barry, Oct 13 2005

Status

approved