# [seqfan] Re: Counting functions for groups

W. Edwin Clark wclark at mail.usf.edu
Fri Oct 14 18:13:42 CEST 2011

```http://oeis.org/A066389  is the number of groups of order n generated by 2
or fewer elements.
But the sequence of groups of order n generated by 3 or fewer elements,
namely,

1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,13,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,44,1,2,1,14,1

is not in the OEIS. (Easy to compute with GAP)

On Fri, Oct 14, 2011 at 1:37 AM, <franktaw at netscape.net> wrote:

> This is not a direct response to David's question. One of the things I try
> to do when I see a proposed sequence is to try to find simpler and/or more
> basic sequences based on the same idea(s).
>
> In this case, combinatorially, a group with distinguished generators is a
> different animal from just the group. Do we have a sequence with the number
> of groups-with-generators with n elements? How about a table of the number
> of groups with n elements and k generators?
>
> In both of these cases, there are actually two sequences: one where the set
> of generators is required to be minimal, and one where it is not (so T(n,n)
> would be A000001(n)).
>
> I do think all four of the sequences I described here do belong in the
> OEIS. I suspect that others here are better placed than I am to compute
> them. (I can't readily even compute enough of them to search and see if they