The Abundancy Index and Feebly Amicable Numbers

TL;DR

This paper generalizes the concept of amicable pairs to feebly amicable pairs based on the abundancy index, investigates their properties, and concludes not all numbers are feebly amicable with others, supported by computational data and conjectures.

Contribution

It introduces and studies feebly amicable pairs, a new generalization of amicable pairs, and provides computational evidence and conjectures about their distribution.

Findings

Not all numbers are feebly amicable with others.

Generated data suggests certain patterns and open questions.

Provides a foundation for further exploration of feebly amicable numbers.

Abstract

This research explores the sum of divisors - $\sigma(n)$ - and the abundancy index given by the function $\frac{\sigma(n)}{n}$. We give a generalization of amicable pairs - feebly amicable pairs (also known as harmonious pairs), that is $m,n$ such that $\frac{n}{\sigma(n)}+ \frac{m}{\sigma(m)}=1$. We first give some groundwork in introductory number theory, then the goal of the paper is to determine if all numbers are feebly amicable with at least one other number by using known results about the abundancy index. We establish that not all numbers are feebly amicable with at least one other number. We generate data using the R programming language and give some questions and conjectures.