Classification of 10 as friendly or solitary

Determine whether 10 is friendly or solitary by deciding whether there exists a positive integer m ≠ 10 whose abundancy index I(m) = σ(m)/m equals 9/5 (the abundancy index of 10), or prove that no such m exists.

Background

The paper studies friends of 10, where two integers are friends if they share the same abundancy index I(n) = σ(n)/n. It is a longstanding problem to decide whether 10 is friendly (has at least one friend) or solitary (has no friend). Prior work has shown structural constraints that any hypothetical friend of 10 must satisfy, such as being an odd square with many distinct prime divisors.

The authors contribute upper bounds for the second, third, and fourth smallest prime divisors of any friend of 10, but the fundamental classification of 10 itself remains unresolved, motivating this explicit open question.