# Superior highly composite number

In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.

For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k larger than 1 we have

$\frac{d(n)}{n^\varepsilon}\geq\frac{d(k)}{k^\varepsilon}$

where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915).

The first superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200... (sequence A002201 in OEIS).

## Properties

All superior highly composite numbers are highly composite.

An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers.[1] Let

$e_p(x) = \left\lfloor \frac{1}{\sqrt[x]{p} - 1} \right\rfloor\quad$

for any prime number p and positive real x. Then

$\quad s(x) = \prod_{p \in \mathbb{P}} p^{e_p(x)}\quad$ is a superior highly composite number.

Note that the product need not be computed indefinitely, because if $p > 2^x$ then $e_p(x) = 0$, so the product to calculate $s(x)$ can be terminated once $p \ge 2^x$.

Also note that in the definition of $e_p(x)$, $1/x$ is analogous to $\varepsilon$ in the implicit definition of a superior highly composite number.

Moreover for each superior highly composite number $s^\prime$ exists a half-open interval $I \subset \R^+$ such that $\forall x \in I: s(x) = s^\prime$ .

This representation implies that there exist an infinite sequence of $\pi_1, \pi_2, \ldots \in \mathbb{P}$ such that for the n-th superior highly composite number $s_n$ holds

$s_n = \prod_{i=1}^n\pi_i$

The first $\pi_i$ are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.