  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.
