Each friend of 10 has at least 10 nonidentical prime factors (Q6559431)

Let \(\sigma(n)\) be the sum of divisors function and \(\sigma_{-1}(n)=\frac{\sigma(n)}{n}\) be the abundancy index function. Two positive integers \(m\neq n\) are \textit{friends} if \(\sigma_{-1}(m)=\sigma_{-1}(n)\) and \(n\) is a \textit{solitary} number if it has no friends.\N\NThe smallest positive integer for which no friends are known and which has not be proven to be solitary is \(10\).\N\NUsing an adaption of the factor-chain-search scheme developed by \textit{P. P. Nielsen} to study odd perfect numbers [Math. Comput. 76, No. 260, 2109--2126 (2007; Zbl 1142.11086)], the author is able to prove that any friend of \(10\), if it exists, must have at least \(10\) distinct prime factors.