A322885 Number of 3-generated Abelian groups of order n.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1

Offset

1,4

Comments

Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A046951(n).

Sum of the first three columns from A249770 (for n > 1).

Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = A001399(e).

Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).

Sum_{k=1..n} a(k) ~ Pi^2*Zeta(3)*n/6 + Zeta(1/2)*Zeta(3/2)*sqrt(n) + Zeta(1/3)*Zeta(2/3)*n^(1/3). - Vaclav Kotesovec, Feb 02 2019

Maple

f:= proc(n) local t;
  mul(round((t[2]+3)^2/12), t=ifactors(n)[2])
end proc:
map(f, [$1..200]); # Robert Israel, May 20 2019

Mathematica

a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 02 2019 *)

Crossrefs

Cf. A001399, A010052, A010057, A046951, A061704, A249770.

Sequence in context: A091050 A005361 A303915 * A292582 A008479 A331178

Adjacent sequences: A322882 A322883 A322884 * A322886 A322887 A322888

Keyword

nonn,mult

Author

Álvar Ibeas, Dec 29 2018

Status

approved