A274847 -id:A274847 - cp's OEIS frontend

A289445 a(n) = number of similarity classes of groups G with exactly n subgroups and such that G = G-tilde, where G-tilde is the unique subgroup of G left after factoring out the cyclic, central Sylow subgroups.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 0, 4, 0, 8, 3, 4

Offset: 1

N. J. A. Sloane, Jul 08 2017

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).

Cf. A274847.

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

A289445 a(n) = number of similarity classes of groups G with exactly n subgroups and such that G = G-tilde, where G-tilde is the unique subgroup of G left after factoring out the cyclic, central Sylow subgroups.

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