Highly composite number

This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers

Definition

A natural number $n$ is termed a highly composite number if it is a strict maximum-so-far for the divisor count function. In other words, if $\tau (n)$ denotes the number of divisors of $n$, then $n$ is highly composite if $\tau (n)>\tau (k)$ for every natural number $k<n$.

Relation with other properties

• Superabundant number is a closely related notion -- it is a strict maximum-so-far for the ratio of the divisor sum function to the number itself.