A034677 - OEIS

%I #31 Sep 14 2020 02:55:58

%S 1,9,28,65,126,252,344,513,730,1134,1332,1820,2198,3096,3528,4097,

%T 4914,6570,6860,8190,9632,11988,12168,14364,15626,19782,19684,22360,

%U 24390,31752,29792,32769,37296,44226,43344,47450,50654,61740,61544,64638,68922,86688,79508

%N Sum of cubes of unitary divisors of n.

%C A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.

%H 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 Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-3). - _R. J. Mathar_, Mar 04 2011

%F If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(3*k_j)). - _Ilya Gutkovskiy_, Nov 04 2018

%F Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (360 * Zeta(5)). - _Vaclav Kotesovec_, Feb 01 2019

%e The unitary divisors of 6 are 1, 2, 3 and 6, so a(6) = 252.

%t scud[n_]:=Total[Select[Divisors[n],CoprimeQ[#,n/#]&]^3]; Array[scud,40] (* _Harvey P. Dale_, Oct 16 2016 *)

%t 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 (PARI) A034677_vec(len)={

%o         a000012=direuler(p=2,len, 1/(1-X)) ;

%o         a000578=direuler(p=2,len, 1/(1-p^3*X)) ;

%o         a000578x=direuler(p=2,len, 1-p^3*X^2) ;

%o         dirmul(dirmul(a000012,a000578),a000578x)

%o }

%o A034677_vec(70) /* via D.g.f., _R. J. Mathar_, Mar 05 2011 */

%Y Cf. A034444, A034448.

%Y Row n=3 of A286880.

%K nonn,mult

%O 1,2

%A _Erich Friedman_