# Aliquot sequences: a soft question about why aren't variants of the aliquot sequence for different divisor functions

I know from an informative point of view the theory of aliquot sequences: definitions, closely-related sequences, and main propositions and theorems.

The first section of the Wikipedia Aliquot sequence explains the definition and overview of the problem.

On the other hand I know that there are more divisor functions $$\sigma_k(n)=\sum_{d\mid n}d^k$$, with $$k\geq 2$$ integer, from which I wondered if one can to propose different aliquot sums, and the corresponding aliquot sequences as variants.

Question. Are known aliquot sequences defined as problems involving in the definition different divisor functions $$\sigma_k(n)$$? Why? Is it because the resulting problems, the corresponding aliquot recurrences, were explored but were not as interesting as the original aliquot sequence for $$\sigma(n)$$? If it is in the literature add the reference and I try to search and read from the literature those variants. Many thanks.

As comparison, I believe that in the literature the Collatz conjecture has different variants with a good mathematical content.

• Many thanks @GerryMyerson – user142929 Sep 10 at 7:49