Does Ten Have a Friend?

TL;DR

The paper investigates the properties of integers with a specific abundancy index, revealing structural constraints such as being a square with multiple prime factors and specific modular conditions on these primes.

Contribution

It establishes new necessary conditions for integers with abundancy index 9/5, including prime factorization and modular constraints, advancing understanding of such numbers.

Findings

Any such integer must be a square with at least 6 prime factors.

One prime factor must be congruent to 1 modulo 3.

That prime factor must have an exponent congruent to 2 modulo 6.

Abstract

Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of $n$.