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 $3\nmid N$, then $\Omega \ge \frac{8}{3}\omega (N)-\frac{7}{3}$ and if $3|N$ then $\Omega (N)\ge \frac{21}{8}\omega (N)-\frac{39}{8}.$

6 pages, accepted to INTEGERS