Sequence of Record Peaks in Values of Divisor Sum

Theorem

Let $d: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:

$\map d n = \map {\sigma_1} n - \map {\sigma_1} {n'}$

where:

$n'$ denotes the previous highly abundant number.

The following $n \in \Z_{>0}$ have the property that they have a higher value of $\map d n$ than any smaller $n$:

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

That is, they are the peak divisor sums which exceed the previous peak by a higher number than any previous peak.

That is, they are highly abundant number which have divisor sums whose difference with that of the divisor sum of the previous highly abundant numbers is greater than that with the previous record difference.

$n$	$n'$	$\map {\sigma_1} n$	$\map {\sigma_1} {n'}$	$\map d n$
$2$	$1$	$3$	$1$	$2$
$4$	$3$	$7$	$4$	$3$
$6$	$4$	$12$	$7$	$5$
$12$	$10$	$28$	$18$	$10$
$24$	$20$	$60$	$42$	$18$
$36$	$30$	$91$	$72$	$19$
$48$	$42$	$124$	$96$	$28$
$60$	$48$	$168$	$124$	$44$
$120$	$108$	$360$	$280$	$80$