# 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.