Can someone show me how to prove that $$\liminf\_{n \to \infty} \frac{\sigma\_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open?
Here, $\sigma\_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of divisors function.


Note: I think for $k=2$ this had been proved by Makowski and Schinzel and the limit equals $1$.


Thank you for any help.



Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?
It is an open problem, only known for $k=1,2$.


Both the conjecture and the known cases are due to Schinzel.


You can find a nice survey here: "[On the third iterates of the φ- and σ-functions](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-3dd8ec9f-5996-4945-a5ee-ac422cd6097a)" H. Maier (1984)


It shouldn't be hard to find more recent references that mention it as open.


