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.

  • 2
    $\begingroup$ @Ricardo Andrade: I changed back $\sigma_k(n)$ to its originally intended meaning. For the sum of $k$-th powers of $n$ it is trivial that the liminf is infinite. $\endgroup$
    – GH from MO
    Jul 21 '15 at 21:30
  • $\begingroup$ @GH from MO, thank you very much for your correction. I apologize for my mistaken description. $\endgroup$ Jul 23 '15 at 20:40

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.

  • 1
    $\begingroup$ and Maier has used sieve methods to prove the result for k = 3. $\endgroup$ Jul 22 '15 at 2:13

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