Hi, Wadim.
nikmil, in your comment an hour ago you should have put \sigma( a_n) as your curly braces just disappear.
So 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 $f_1$ 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.