TITLE: Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?
QUESTION [6 upvotes]: 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.

REPLY [8 votes]: 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" H. Maier (1984)
It shouldn't be hard to find more recent references that mention it as open.