Sequence of Record Peaks in Values of Sigma
Let $d: \Z_{>0} \to \Z_{>0}$ be the mapping defined as: :$d \left({n}\right) = \sigma \left({n}\right) - \sigma \left({n'}\right)$ 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 $d \left({n}\right)$ than any smaller $n$: :$\forall m \in \Z_{>0}: m < n \implies \sigma \left({m}\right) < \sigma \left({n}\right)$ 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" | $\sigma \left({n}\right)$  ! align="right" style = "padding: 2px 10px" | $\sigma \left({n'}\right)$  ! 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" | $80$ |}