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%I #13 Jul 15 2020 12:36:39
%S 1,1,1,2,2,3,1,5,2,5,2,5,1,6,4,9,2,7,1,11,2,6,3,11,3,8,4,9,3,14,1,16,
%T 3,6,4,15,2,8,2,21,2,13,2,13,8,6,2,23,4
%N a(n) is the number of similarity classes of abelian groups with exactly n subgroups (see reference for precise definition of similarity classes).
%C See Slattery references for a precise definition of similarity.
%C See Betz and Nash first reference for proof of the first 22 terms.
%C See Betz and Nash second reference for proof of terms 23--49.
%H Alexander Betz and David A. Nash, <a href="https://arxiv.org/abs/2006.11315">Classifying groups with a small number of subgroups</a>, arXiv:2006.11315 [math.GR], (2020).
%H Alexander Betz and David A. Nash, <a href="https://www.researchgate.net/publication/342918790_A_note_on_abelian_groups_with_fewer_than_50_subgroups">A note on abelian groups with fewer than 50 subgroups</a>, preprint, (2020).
%H G. A. Miller, <a href="http://www.pnas.org/content/25/7/367.full.pdf">Groups having a small number of subgroups</a>, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
%H M. C. Slattery, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.1.78">On a property motivated by groups with a specified number of subgroups</a>, Amer. Math. Monthly, 123 (2016), 78-81.
%H M. C. Slattery, <a href="http://arxiv.org/abs/1607.01834">Groups with at most twelve subgroups</a>, arXiv:1607.01834 [math.GR], 2016-2020.
%e For n = 6, a(6) = 3 and the three similarity classes of abelian groups with exactly six subgroups are Z_{p^5}, Z_{p^2q}, and Z_3 X Z_3.
%Y Cf. A274847, A018216, A289445.
%K nonn,more
%O 1,4
%A _David A. Nash_, Jun 29 2020
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