An improvement of an inequality of Ochem and Rao concerning odd perfect numbers

TL;DR

This paper improves existing inequalities relating the total and distinct prime divisors of odd perfect numbers, providing tighter bounds depending on divisibility by 3.

Contribution

It presents improved inequalities connecting the total and distinct prime divisors of odd perfect numbers, refining previous bounds based on divisibility by 3.

Findings

If 3 does not divide N, then Ω ≥ (8/3)ω - (7/3).

If 3 divides N, then Ω ≥ (21/8)ω - (39/8).

Provides sharper bounds for the prime divisor counts of odd perfect numbers.

Abstract

Let $\Omega(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $\omega(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $\omega(N)$ and $\Omega(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $\Omega \geq \frac{8}{3}\omega(N)-\frac{7}{3}$ and if $3 |N$ then $\Omega(N) \geq \frac{21}{8}\omega(N)-\frac{39}{8}.$