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.

Occurrence

Initial values

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, [SHOW MORE]

840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160

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