# Superior highly composite number

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Divisor function d(n) up to n = 250

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.

The first 10 superior highly composite numbers and their factorization are listed.

Order SHCN
n
prime
factorization
prime
exponents
prime
factors
d(n) primorial
factorization
1 2 2 1 1 2 2
2 6 2 ⋅ 3 1,1 2 4 6
3 12 22 ⋅ 3 2,1 3 6 2 ⋅ 6
4 60 22 ⋅ 3 ⋅ 5 2,1,1 4 12 2 ⋅ 30
5 120 23 ⋅ 3 ⋅ 5 3,1,1 5 16 22 ⋅ 30
6 360 23 ⋅ 32 ⋅ 5 3,2,1 6 24 2 ⋅ 6 ⋅ 30
7 2520 23 ⋅ 32 ⋅ 5 ⋅ 7 3,2,1,1 7 48 2 ⋅ 6 ⋅ 210
8 5040 24 ⋅ 32 ⋅ 5 ⋅ 7 4,2,1,1 8 60 22 ⋅ 6 ⋅ 210
9 55440 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 4,2,1,1,1 9 120 22 ⋅ 6 ⋅ 2310
10 720720 24 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 4,2,1,1,1,1 10 240 22 ⋅ 6 ⋅ 30030

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

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

and for all natural numbers k larger than n we have

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

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

The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors.

## 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

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

for any prime number p and positive real x. Then

${\displaystyle \quad s(x)=\prod _{p\in \mathbb {P} }p^{e_{p}(x)}\quad }$ is a superior highly composite number.

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

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

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

This representation implies that there exist an infinite sequence of ${\displaystyle \pi _{1},\pi _{2},\ldots \in \mathbb {P} }$ such that for the n-th superior highly composite number ${\displaystyle s_{n}}$ holds

${\displaystyle s_{n}=\prod _{i=1}^{n}\pi _{i}}$

The first ${\displaystyle \pi _{i}}$ 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.

## Notes

1. Ramanujan (1915); see also URL http://wwwhomes.uni-bielefeld.de/achim/hcn.dvi

## References

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