Definition:Highly Abundant Number

Definition
Let $n \in \Z_{>0}$ be a positive integer.

Then $n$ is highly abundant :
 * $\forall m \in \Z_{>0}, m < n: \map \sigma m < \map \sigma n$

where $\map \sigma n$ is the $\sigma$ function of $n$.

That is, $n$ has a higher $\sigma$ value than any smaller positive integer.

Also defined as
Some sources use the term highly abundant number for what is defined on as highly composite number.

Also see

 * Definition:Superabundant Number