Generalizing the Abundancy of an Integer

Abstract

The abundancy index of a positive integer is the ratio between the sum of its divisors and itself. We generalize previous results on abundancy indices by defining a two-variable abundancy index function as $I(x,n):{\mathbb{Z}}^{+}\times {\mathbb{Z}}^{+}\to \mathbb{Q}$ where $I(x,n)=\frac{{\sigma }_x(n)}{n^x}$. Specifically, we extend limiting properties of the abundancy index and construct sufficient conditions for rationals greater than one that fail to be in the image of the function $I(x,n)$.