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A034677 Sum of cubes of unitary divisors of n.

%I A034677 #31 Sep 14 2020 02:55:58
%S A034677 1,9,28,65,126,252,344,513,730,1134,1332,1820,2198,3096,3528,4097,
%T A034677 4914,6570,6860,8190,9632,11988,12168,14364,15626,19782,19684,22360,
%U A034677 24390,31752,29792,32769,37296,44226,43344,47450,50654,61740,61544,64638,68922,86688,79508
%N A034677 Sum of cubes of unitary divisors of n.
%C A034677 A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.
%H A034677 Amiram Eldar, <a href="/A034677/b034677.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%F A034677 Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-3). - _R. J. Mathar_, Mar 04 2011
%F A034677 If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(3*k_j)). - _Ilya Gutkovskiy_, Nov 04 2018
%F A034677 Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (360 * Zeta(5)). - _Vaclav Kotesovec_, Feb 01 2019
%e A034677 The unitary divisors of 6 are 1, 2, 3 and 6, so a(6) = 252.
%t A034677 scud[n_]:=Total[Select[Divisors[n],CoprimeQ[#,n/#]&]^3]; Array[scud,40] (* _Harvey P. Dale_, Oct 16 2016 *)
%t A034677 f[p_, e_] := p^(3*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)
%o A034677 (PARI) A034677_vec(len)={
%o A034677         a000012=direuler(p=2,len, 1/(1-X)) ;
%o A034677         a000578=direuler(p=2,len, 1/(1-p^3*X)) ;
%o A034677         a000578x=direuler(p=2,len, 1-p^3*X^2) ;
%o A034677         dirmul(dirmul(a000012,a000578),a000578x)
%o A034677 }
%o A034677 A034677_vec(70) /* via D.g.f., _R. J. Mathar_, Mar 05 2011 */
%Y A034677 Cf. A034444, A034448.
%Y A034677 Row n=3 of A286880.
%K A034677 nonn,mult
%O A034677 1,2
%A A034677 _Erich Friedman_