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arXiv · 1706.07009

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

Joshua Zelinsky
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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 $\Omega(N) \geq \frac{21}{8}\omega(N)-\frac{39}{8}.$

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Joshua Zelinsky. 2017-06-21. An improvement of an inequality of Ochem and Rao concerning odd perfect numbers. https://arxiv.org/abs/1706.07009

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