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A034382
Number of labeled Abelian groups of order n.
7
1, 2, 3, 16, 30, 360, 840, 15360, 68040, 907200, 3991680, 159667200, 518918400, 14529715200, 163459296000, 4250979532800, 22230464256000, 1200445069824000, 6758061133824000, 405483668029440000
OFFSET
1,2
LINKS
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.
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).
CROSSREFS
KEYWORD
nonn
EXTENSIONS
a(16) corrected by Max Alekseyev, Sep 12 2019
STATUS
approved