A note on odd perfect numbers

TL;DR

This paper investigates properties of odd perfect numbers, establishing bounds on their divisors and the sum-of-divisors function, contributing to the understanding of their structure and limitations.

Contribution

It proves that for an odd perfect number, the ratio of the sum-of-divisors of a reduced number to a prime power divisor exceeds 5, and shows bounds on odd perfect numbers based on divisor sum ratios.

Findings

If $N$ is an odd perfect number, then $rac{\sigma(N/q^{ ext{ extalpha}})}{q^{ ext{ extalpha}}}>5$.

Bounds on $N$ depend on the ratio $rac{\sigma(N/q^{ ext{ extalpha}})}{q^{ ext{ extalpha}}}$.

The ratio of sum-of-divisors to prime power divisor influences the size constraints of odd perfect numbers.

Abstract

In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if $\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.