Except the original Grönwall's theorem that $$\limsup\_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $$\limsup\_{\begin{smallmatrix} n\to\infty\cr n\ \text{is square free}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{6e^{\gamma}}{\pi^2}$$ and $$\limsup\_{\begin{smallmatrix} n\to\infty\cr n\ \text{is odd}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{e^{\gamma}}{2}$$ that have been proven [here](http://www.mpim-bonn.mpg.de/preprints/send?bid=2960), are there any other similar statements known?



Variants of Grönwall's theorem
One example possessing a limit is the colossally abundant numbers of Alaoglu and Erdos,  

<http://en.wikipedia.org/wiki/Colossally_abundant_number> 


where the limit of the Choie, Lichiardopol, Moree and Sole's 
 $$f\_1(a\_n) = \frac{\sigma(a\_n)}{a\_n \log \log a\_n}$$
 is the same
$$ e^\gamma .$$
That is, the limit for these numbers is the lim sup for all numbers.


These are more natural than people realize. There is a simple recipe that takes some $ \epsilon > 0$ and gives an explicit factorization for the best value $n\_\epsilon;$ see page 7 in the Briggs pdf
"Notes on the Riemann hypothesis and abundant numbers" at the bottom of the Wikipedia entry. The exponent of a prime $p$ in the factorization of $n\_\epsilon$ is
$$ \left\lfloor \log\_p \left( \frac{p^{1 + \epsilon} - 1}{p^\epsilon -1} \right) \right\rfloor - 1 $$


The process of making a sequence of "champion" numbers this way was invented by Ramanujan.


