Definition:Highly Abundant Number

Definition

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

Then $n$ is highly abundant if and only if:

$\forall m \in \Z_{>0}, m < n: \map {\sigma_1} m < \map {\sigma_1} n$

where $\sigma_1$ denotes the divisor sum function of $n$.

That is, if and only if $n$ has a higher divisor sum than any smaller positive integer.

$1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, \ldots$

The divisor sum values of the first few highly abundant numbers can be tabulated as:

$n$	$\map {\sigma_1} n$	Difference from previous peak
$1$	$1$
$2$	$3$	$2$
$3$	$4$	$1$
$4$	$7$	$3$
$6$	$12$	$5$
$8$	$15$	$3$
$10$	$18$	$3$
$12$	$28$	$10$
$16$	$31$	$3$
$18$	$39$	$8$
$20$	$42$	$3$
$24$	$60$	$18$
$30$	$72$	$12$
$36$	$91$	$19$
$42$	$96$	$5$
$48$	$124$	$28$
$60$	$168$	$44$
$72$	$195$	$27$
$84$	$224$	$29$
$90$	$234$	$10$
$96$	$252$	$18$
$108$	$280$	$28$
$120$	$360$	$80$

Thus the sequence of peak divisor sums begins:

$1, 3, 4, 7, 12, 15, 18, 28, 31, 39, 42, 60, 72, 91, 96, 124, 168, 195, 224, 234, 252, 280, 360, \ldots$

Some sources use the term highly abundant number for what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as highly composite number.