A034382 Number of labeled Abelian groups of order n.

1, 2, 3, 16, 30, 360, 840, 15360, 68040, 907200, 3991680, 159667200, 518918400, 14529715200, 163459296000, 4250979532800, 22230464256000, 1200445069824000, 6758061133824000, 405483668029440000

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• Hy Ginsberg, Totally Symmetric Quasigroups of Order 16, arXiv:2211.13204 [math.CO], 2022.
• C. J. Hillar and D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR], 2006.
• Sugarknri et al., Number of labeled Abelian groups of order n, Mathematics Stack Exchange, 2019.
• Index entries for sequences related to groups

Crossrefs

Cf. A000688, A034381, A034383, A058159.

Formula

a(n) = A058162(n) * n.

a(n) = Sum n!/|Aut(G)|, where the sum is taken over the different products G of cyclic groups with |G|=n. Formula for |Aut(G)| is given by Hillar and Rhea (2007). Another formula is given by Sugarknri (2019).

Extensions

a(16) corrected by Max Alekseyev, Sep 12 2019