Sequence of Record Peaks in Values of Divisor Sum

Theorem
Let $d: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:
 * $\map d n = \map \sigma n - \map \sigma {n'}$

where:
 * $n$ denotes a highly abundant number
 * $n'$ 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 $n$:
 * $\forall m \in \Z_{>0}: m < n \implies \map \sigma m < \map \sigma n$

That is, they are the peak $\sigma$ values which exceed the previous peak by a higher number than any previous peak.

That is, they are highly abundant number which have $\sigma$ values whose difference with that of the $\sigma$ (sigma) value of the previous highly abundant numbers is greater than that with the previous record difference.


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $n'$ ! align="right" style = "padding: 2px 10px" | $\map \sigma n$ ! align="right" style = "padding: 2px 10px" | $\map \sigma {n'}$ ! style = "padding: 2px 10px" | $d \left({n}\right)$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $28$
 * align="right" style = "padding: 2px 10px" | $18$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $24$
 * align="right" style = "padding: 2px 10px" | $20$
 * align="right" style = "padding: 2px 10px" | $60$
 * align="right" style = "padding: 2px 10px" | $42$
 * align="right" style = "padding: 2px 10px" | $18$
 * align="right" style = "padding: 2px 10px" | $36$
 * align="right" style = "padding: 2px 10px" | $30$
 * align="right" style = "padding: 2px 10px" | $91$
 * align="right" style = "padding: 2px 10px" | $72$
 * align="right" style = "padding: 2px 10px" | $19$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $42$
 * align="right" style = "padding: 2px 10px" | $124$
 * align="right" style = "padding: 2px 10px" | $96$
 * align="right" style = "padding: 2px 10px" | $28$
 * align="right" style = "padding: 2px 10px" | $60$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $168$
 * align="right" style = "padding: 2px 10px" | $124$
 * align="right" style = "padding: 2px 10px" | $44$
 * align="right" style = "padding: 2px 10px" | $120$
 * align="right" style = "padding: 2px 10px" | $108$
 * align="right" style = "padding: 2px 10px" | $360$
 * align="right" style = "padding: 2px 10px" | $280$
 * align="right" style = "padding: 2px 10px" | $80$
 * }
 * align="right" style = "padding: 2px 10px" | $124$
 * align="right" style = "padding: 2px 10px" | $44$
 * align="right" style = "padding: 2px 10px" | $120$
 * align="right" style = "padding: 2px 10px" | $108$
 * align="right" style = "padding: 2px 10px" | $360$
 * align="right" style = "padding: 2px 10px" | $280$
 * align="right" style = "padding: 2px 10px" | $80$
 * }
 * }