A Criterion for Deficient Numbers Using the Abundancy Index and Deficiency Functions

TL;DR

The paper establishes a new criterion for identifying almost perfect numbers using the abundancy index and deficiency functions, providing a mathematical condition that characterizes these numbers precisely.

Contribution

It introduces a novel criterion linking the abundancy index and deficiency functions to characterize almost perfect numbers, extending to integers with deficiency greater than one.

Findings

A necessary and sufficient condition for almost perfect numbers involving I(n) and D(n)

Extension of the criterion to integers with D(m) > 1

Mathematical characterization of number properties using index and deficiency functions

Abstract

We show that $n$ is almost perfect if and only if $I(n) - 1 < D(n) \leq I(n)$, where $I(n)$ is the abundancy index of $n$ and $D(n)$ is the deficiency of $n$. This criterion is then extended to the case of integers $m$ satisfying $D(m)>1$.