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Sum of cube divisors of n.
11

%I #31 Dec 01 2020 02:59:05

%S 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,

%T 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,

%U 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

%N Sum of cube divisors of n.

%C 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

%H Antti Karttunen, <a href="/A113061/b113061.txt">Table of n, a(n) for n = 1..19683</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a>, arXiv:1106.4038 [math.NT] (2011), Remark 15.

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)). - _Ilya Gutkovskiy_, Dec 24 2016

%F a(n) = Sum_{d|n} A010057(d)*d. - _Antti Karttunen_, Oct 08 2017

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

%p A113061 := proc(n)

%p local a,pe,p,e;

%p a := 1;

%p for pe in ifactors(n)[2] do

%p p := pe[1] ;

%p e := pe[2] ;

%p e := 3*(1+floor(e/3)) ;

%p a := a*(p^e-1)/(p^3-1) ;

%p end do:

%p a ;

%p end proc:

%p seq(A113061(n),n=1..100) ; # _R. J. Mathar_, Oct 08 2017

%t a[n_] := Sum[If[IntegerQ[d^(1/3)], d, 0], {d, Divisors[n]}];

%t Array[a, 100] (* _Jean-François Alcover_, Nov 25 2017 *)

%t 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 *)

%o (PARI) A113061(n) = sumdiv(n,d,ispower(d,3)*d); \\ _Antti Karttunen_, Oct 08 2017

%o (Scheme)

%o ;; With memoization-macro definec, after the multiplicative formula of _R. J. Mathar_:

%o (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

%Y Cf. A004709, A010057, A035316.

%K nonn,mult

%O 1,8

%A _Paul Barry_, Oct 13 2005