Definition:Highly Composite Number

Definition

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

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

$\forall m \in \Z_{>0}, m < n: \map {\sigma_0} m < \map {\sigma_0} n$

where $\map {\sigma_0} n$ is the divisor count 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. https://mathworld.wolfram.com/HighlyCompositeNumber.html