cp's OEIS Frontend

A064803 Number of subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

%I A064803 #26 Jan 07 2025 20:26:09
%S A064803 1,16,28,129,64,448,116,802,445,1024,268,3612,368,1856,1792,4387,616,
%T A064803 7120,764,8256,3248,4288,1108,22456,2607,5888,5776,14964,1744,28672,
%U A064803 1988,22308,7504,9856,7424,57405,2816,12224,10304,51328,3448,51968,3788,34572,28480,17728,4516,122836,9009,41712
%N A064803 Number of subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
%H A064803 Amiram Eldar, <a href="/A064803/b064803.txt">Table of n, a(n) for n = 1..10000</a>
%H A064803 Mario Hampejs and László Tóth, <a href="http://ac.inf.elte.hu/Vol_039_2013/111_39.pdf">On the subgroups of finite Abelian groups of rank three</a>, Annales Univ. Sci. Budapest., Sect. Comp. 39 (2013) 111-124.
%F A064803 For a prime p: a(p) = 2*(p^2+p+2). - _Vladeta Jovovic_, Oct 22 2001
%F A064803 Multiplicative with a(p^e) = Sum_{j=0..2*e} (e - floor((j - 1)/2))*(2*j - floor((j - 1)/2))*p^(2*e - j). - _Amiram Eldar_, Nov 29 2022
%p A064803 A064803 := proc(n)
%p A064803     local a,f,nu,p,j ;
%p A064803     a := 1 ;
%p A064803     for f in ifactors(n)[2] do
%p A064803         nu := op(2,f) ;
%p A064803         p := op(1,f) ;
%p A064803         add( (nu-floor((j-1)/2))*(2*j-floor((j-1)/2))*p^(2*nu-j),j=0..2*nu) ;
%p A064803         a := a*% ;
%p A064803     end do:
%p A064803     a ;
%p A064803 end proc: # _R. J. Mathar_, May 11 2013
%t A064803 f[p_, e_] := Sum[(e - Floor[(j - 1)/2])*(2*j - Floor[(j - 1)/2])*p^(2*e - j), {j, 0, 2*e}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 29 2022 *)
%Y A064803 Cf. A060648, A060724.
%K A064803 nonn,mult
%O A064803 1,2
%A A064803 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 21 2001
%E A064803 More terms from _Laszlo Toth_, May 11 2013