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

Preprint English OPEN
Zelinsky, Joshua (2017)
  • Subject: 11A05 (Primary), 11A25 (Secondary) | Mathematics - Number Theory

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 $\Omega(N) \geq \frac{21}{8}\omega(N)-\frac{39}{8}.$
  • References (2)

    [1] On the number of prime factors of an odd perfect number. Pascal Ochem and Michael Rao. Math. Comp. 83 (2014), 2435-2439.

    [2] Odd perfect numbers, Diophantine equations, and upper bounds. P.P. Nielsen. Math. Comp. 84 (2015) 2549-2567.

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