Generalizing the Abundancy of an Integer

TL;DR

This paper extends the concept of the abundancy index to a two-variable function, exploring its properties and identifying conditions under which certain rationals are not in its image.

Contribution

It introduces a generalized two-variable abundancy index function and analyzes its limiting behavior and image, providing new insights into number theory.

Findings

Extended the abundancy index to two variables.

Identified conditions for rationals not in the image.

Analyzed limiting properties of the generalized index.

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 $I(x,n)\colon\mathbb{Z^+}\times\mathbb{Z^+}\to\mathbb{Q}$ 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 $I(x,n)$.