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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 ofsome positive integer m equals the abundancy index of n but m is not equal ton, then m and n are called friends. A positive integer with no friends iscalled solitary. The smallest positive integer that is not known to have afriend and is not known to be solitary is 10.  It is not known if the number 6 has odd friends, that is, if odd perfectnumbers exist. In a 2007 article, Nielsen proved that the number ofnonidentical prime factors in any odd perfect number is at least 9. A 2015article by Nielsen, which was more complicated and used a computer program thattook months to complete, increased the lower bound from 9 to 10.  This work applies methods from Nielsen's 2007 article to show that eachfriend of 10 has at least 10 nonidentical prime factors.  This is a formal write-up of results to be presented at the Southern AfricaMathematical Sciences Association Conference 2023 at the University ofPretoria.

Each friend of 10 has at least 10 nonidentical prime factors