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

The Abundancy Index and Feebly Amicable Numbers