Each friend of 10 has at least 10 nonidentical prime factors

For each positive integer $n$, if the sum of the factors of $n$ is divided by $n$, then the result is called the abundancy index of $n$. If the abundancy index of some positive integer $m$ equals the abundancy index of $n$ but $m$ is not equal to $n$, then $m$ and $n$ are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.

It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.

This work applies methods from Nielsen’s 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.

This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria.