# Definition:Highly Composite Number

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## Definition

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

Then $n$ is **highly composite** if and only if:

- $\forall m \in \Z_{>0}, m < n: \map \tau m < \map \tau n$

where $\map \tau n$ is the divisor counting (tau) function of $n$.

That is, if and only if $n$ has a larger number of divisors than any smaller positive integer.

### Sequence of Highly Composite Numbers

The sequence of highly composite numbers begins:

- $1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, \ldots$

## Examples

### $1$ is Highly Composite

$1$ is a highly composite number, being the smallest positive integer with $1$ divisor or more.

### $2$ is Highly Composite

$2$ is a highly composite number, being the smallest positive integer with $2$ divisors or more.

### $60$ is Highly Composite

$60$ is a highly composite number, being the smallest positive integer with $12$ divisors or more.

## Also known as

Some sources use the term **highly abundant number**, but $\mathsf{Pr} \infty \mathsf{fWiki}$ uses that term for a different concept.

## Also see

- Results about
**highly composite numbers**can be found here.

## Sources

- 1927: G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson:
*Collected Papers of Srinivasa Ramanujan* - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $60$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $60$

- Weisstein, Eric W. "Highly Composite Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/HighlyCompositeNumber.html