On the quantity $I(q^k) + I(n^2)$ where $q^k n^2$ is an odd perfect number -- Part II

TL;DR

This paper investigates bounds on the sum of abundancy indices related to odd perfect numbers, aiming to improve lower bounds and derive an upper bound for the prime exponent in such numbers.

Contribution

It introduces an improved lower bound for the sum of abundancy indices in odd perfect numbers and provides an upper bound for the prime exponent $k$.

Findings

Established a new lower bound for $I(q^k) + I(n^2)$

Derived an upper bound for the prime exponent $k$

Extended previous approaches to refine bounds on odd perfect numbers

Abstract

In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and $I(x)$ is the abundancy index of the positive integer $x$. In particular, this yields an upper bound for $k$.