A137315 related to A139795 ?

[Benoît Jubin]

Indeed,  A137315(n)>=A139795(n), but the inequality can be strict. I
sent the following comment and reference a few days ago:

%C A137315 The case of the cyclic groups shows that a(n)>=A139795(n).
This inequality can be strict: if M denotes the Mathieu group M_{22}
of order 2^7.3^2.5.7.11, then Aut(12.M) = M.2, so that
a(2^8.3^2.5.7.11 + 1) > 2^9.3^3.5.7.11, but A139795(2^8.3^2.5.7.11
+ 1) = 2.3.5.7^2.11.13.23 + 1 < 2^9.3^3.5.7.11.
%D A137315 John N. Bray, Robert A. Wilson, On the orders of
automorphism groups of finite groups, Bull. London Math. Soc. 37
(2005) 381--385.

Our ignorance about A137315 can be seen in the huge gap between this
lower bound and the upper bound ~n^(n + n ln n).

Benoit



On Sat, May 31, 2008 at 11:02 AM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
> On Sat, May 31, 2008 at 1:13 PM, Richard Mathar
> <mathar at strw.leidenuniv.nl> wrote:
>> http://research.att.com/~njas/sequences/?q=id:A139795|id:A137315
>> one would think they are related. It might be useful to add comments on
>> whether this is some coincidence or just revealing that they are duplicates.
>
> Certainly not a coincidence - this seems to be a consequence of
>
> Lemma 9. Cn , the cyclic group of order n, has precisely n
> endomorphisms and precisely
> phi(n) automorphisms, where phi is the Euler phi-function.
>
> of the MacHale-Sheehy paper, http://www.ria.ie/cgi-bin/ria/papers/100474.pdf
>
> Maximilian
>