Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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