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%I #29 Nov 25 2022 07:23:47
%S 1,2,3,16,30,360,840,15360,68040,907200,3991680,159667200,518918400,
%T 14529715200,163459296000,4250979532800,22230464256000,
%U 1200445069824000,6758061133824000,405483668029440000
%N Number of labeled Abelian groups of order n.
%H Max Alekseyev, <a href="/A034382/b034382.txt">Table of n, a(n) for n = 1..100</a>
%H Hy Ginsberg, <a href="https://arxiv.org/abs/2211.13204">Totally Symmetric Quasigroups of Order 16</a>, arXiv:2211.13204 [math.CO], 2022.
%H C. J. Hillar and D. Rhea. <a href="https://www.jstor.org/stable/27642365">Automorphisms of finite Abelian groups</a>. American Mathematical Monthly 114:10 (2007), 917-923. Preprint <a href="https://arxiv.org/abs/math/0605185">arXiv:math/0605185</a> [math.GR], 2006.
%H Sugarknri et al., <a href="https://math.stackexchange.com/q/3355137">Number of labeled Abelian groups of order n</a>, Mathematics Stack Exchange, 2019.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F a(n) = A058162(n) * n.
%F 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).
%Y Cf. A000688, A034381, A034383, A058159.
%K nonn
%O 1,2
%A _Christian G. Bower_
%E a(16) corrected by _Max Alekseyev_, Sep 12 2019
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