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A113061 Sum of cube divisors of n.

%I A113061 #31 Dec 01 2020 02:59:05
%S A113061 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 A113061 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 A113061 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 A113061 Sum of cube divisors of n.
%C A113061 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 A113061 Antti Karttunen, <a href="/A113061/b113061.txt">Table of n, a(n) for n = 1..19683</a>
%H A113061 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 A113061 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A113061 G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)). - _Ilya Gutkovskiy_, Dec 24 2016
%F A113061 a(n) = Sum_{d|n} A010057(d)*d. - _Antti Karttunen_, Oct 08 2017
%F A113061 Sum_{k=1..n} a(k) ~ zeta(4/3)*n^(4/3)/4 - n/2. - _Vaclav Kotesovec_, Dec 01 2020
%p A113061 A113061 := proc(n)
%p A113061     local a,pe,p,e;
%p A113061     a := 1;
%p A113061     for pe in ifactors(n)[2] do
%p A113061         p := pe[1] ;
%p A113061         e := pe[2] ;
%p A113061         e := 3*(1+floor(e/3)) ;
%p A113061         a := a*(p^e-1)/(p^3-1) ;
%p A113061     end do:
%p A113061     a ;
%p A113061 end proc:
%p A113061 seq(A113061(n),n=1..100) ; # _R. J. Mathar_, Oct 08 2017
%t A113061 a[n_] := Sum[If[IntegerQ[d^(1/3)], d, 0], {d, Divisors[n]}];
%t A113061 Array[a, 100] (* _Jean-François Alcover_, Nov 25 2017 *)
%t A113061 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 A113061 (PARI) A113061(n) = sumdiv(n,d,ispower(d,3)*d); \\ _Antti Karttunen_, Oct 08 2017
%o A113061 (Scheme)
%o A113061 ;; With memoization-macro definec, after the multiplicative formula of _R. J. Mathar_:
%o A113061 (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 A113061 Cf. A004709, A010057, A035316.
%K A113061 nonn,mult
%O A113061 1,8
%A A113061 _Paul Barry_, Oct 13 2005