A289445 -id:A289445 - cp's OEIS frontend

A274847 a(n) = number of similarity classes of groups with exactly n subgroups (see reference for precise definition of similarity classes).

1, 1, 1, 2, 2, 5, 1, 7, 2, 12, 4, 11, 1, 17, 8, 22, 3, 22, 5

Offset: 1

Michael C Slattery, Jul 08 2016

See Slatterly references for precise definition of similarity classes and a proof of the first 12 terms.

See Betz and Nash for correction of a(10) and proof of terms 13-19.

			For n = 6 the a(6) = 5 similarity classes of groups with 6 subgroups are Z_{p^5}, Z_p X Z_{q^2}, Z_3 X Z_3, S_3, Q_8.

Alexander Betz and David A. Nash, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
Angsuman Das and Arnab Mandal, Solvability of a group based on its number of subgroups, arXiv:2403.01262 [math.GR], 2024.
George A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
David A. Nash and Alexander Betz, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
Michael C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, 123 (2016), 78-81.
Michael C. Slattery, Groups with at most twelve subgroups, arXiv:1607.01834 [math.GR], 2016.

Cf. A018216, A289445.

Correction of a(10) and extension to 19 terms by David A. Nash, Jun 29 2020

A335917 a(n) is the number of similarity classes of abelian groups with exactly n subgroups (see reference for precise definition of similarity classes).

Original entry on oeis.org

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, 3, 6, 4, 15, 2, 8, 2, 21, 2, 13, 2, 13, 8, 6, 2, 23, 4

Offset: 1

David A. Nash, Jun 29 2020

See Slattery references for a precise definition of similarity.
See Betz and Nash first reference for proof of the first 22 terms.
See Betz and Nash second reference for proof of terms 23--49.

			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.

Alexander Betz and David A. Nash, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], (2020).
Alexander Betz and David A. Nash, A note on abelian groups with fewer than 50 subgroups, preprint, (2020).
G. A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
M. C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, 123 (2016), 78-81.
M. C. Slattery, Groups with at most twelve subgroups, arXiv:1607.01834 [math.GR], 2016-2020.

Cf. A274847, A018216, A289445.

cp's OEIS Frontend

A274847 a(n) = number of similarity classes of groups with exactly n subgroups (see reference for precise definition of similarity classes).

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