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

Author

Álvar Ibeas, Dec 29 2018

Keywords

• nonn
• mult
• easy

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.

The number of unordered factorizations of n into biquadratefree power of primes (1 and primes, squares of primes and cubes of primes, A087797). - Amiram Eldar, Jun 12 2025

Links

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

Crossrefs

Cf. A001399, A010052, A010057, A046951, A061704, A087797, A249770.

Programs

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 *)

PARI

a(n) = vecprod(apply(x -> round((x+3)^2/12), factor(n)[, 2])); \\ Amiram Eldar, Jun 12 2025

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