The Abundancy Index of Divisors of Odd Perfect Numbers - Part II

Keneth Adrian P. Dagal, Jose Arnaldo B. Dris

Published 2013-09-04, updated 2015-02-11Version 8

A positive integer $M$ is said to be almost perfect if $\sigma(M) = 2M - 1$, where $\sigma(x)$ is the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. In this article, we show that $q^k$, $n$, $qn$ and $n^2$ are not almost perfect. This then implies that the following inequalities hold: $$I(q) \leq \frac{2(q - 2)}{q},$$ $$I(n) \leq \frac{2n - 2}{n},$$ $$I(qn) \leq \frac{2qn - 2}{qn},$$ and $$I(n^2) \leq \frac{2n^2 - 253}{n^2},$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. Lastly, we show that $$I(n^2) < \frac{2n}{n + 1}$$ if and only if $q < n$.