An improvement of an inequality of Ochem and Rao concerning 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 $\Omega(N) \geq \frac{21}{8}\omega(N)-\frac{39}{8}.$