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 $\tau$ 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