Abstract:

The abundancy index of a positive integer $n$ is defined to be the rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of divisors function $\sigma(n)=\sum_{d\vert n}d$. An abundancy outlaw is a rational number greater than 1 that fails to be in the image of of the map $I$. In this paper, we consider rational numbers of the form $(\sigma(N)+t)/N$ and prove that under certain conditions such rationals are abundancy outlaws.