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).

1, 16, 28, 129, 64, 448, 116, 802, 445, 1024, 268, 3612, 368, 1856, 1792, 4387, 616, 7120, 764, 8256, 3248, 4288, 1108, 22456, 2607, 5888, 5776, 14964, 1744, 28672, 1988, 22308, 7504, 9856, 7424, 57405, 2816, 12224, 10304, 51328, 3448, 51968, 3788, 34572, 28480, 17728, 4516, 122836, 9009, 41712

Offset: 1

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 21 2001

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Mario Hampejs and László Tóth, On the subgroups of finite Abelian groups of rank three, Annales Univ. Sci. Budapest., Sect. Comp. 39 (2013) 111-124.

Crossrefs

Cf. A060648, A060724.

Programs

Maple

A064803 := proc(n)
    local a,f,nu,p,j ;
    a := 1 ;
    for f in ifactors(n)[2] do
        nu := op(2,f) ;
        p := op(1,f) ;
        add( (nu-floor((j-1)/2))*(2*j-floor((j-1)/2))*p^(2*nu-j),j=0..2*nu) ;
        a := a*% ;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2013

Mathematica

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 *)

Formula

For a prime p: a(p) = 2*(p^2+p+2). - Vladeta Jovovic, Oct 22 2001

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

Extensions

More terms from Laszlo Toth, May 11 2013