\begin{definition}[Definition:Highly Abundant Number]
Let $n \in \Z_{>0}$ be a positive integer.
Then $n$ is '''highly abundant''' {{iff}}:
:$\forall m \in \Z_{>0}, m < n: \map {\sigma_1} m < \map {\sigma_1} n$
where $\sigma_1$ denotes the divisor sum function of $n$.
That is, {{iff}} $n$ has a higher divisor sum than any smaller positive integer.
\end{definition}