Definition:Highly Abundant Number

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Definition

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

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

$\forall m \in \Z_{>0}, m < n: \sigma \left({m}\right) < \sigma \left({n}\right)$

where $\sigma \left({n}\right)$ is the $\sigma$ function of $n$.


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


Sequence

The sequence of highly abundant numbers begins:

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


Examples

The $\sigma$ (sigma) values of the first few highly abundant numbers can be tabulated as:

$n$ $\sigma \left({n}\right)$ 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 $\sigma$ values begins:

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


Also defined as

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


Also see

  • Results about highly abundant numbers can be found here.