Superior highly composite number

In mathematics, a superior highly composite number is a natural number n for which exists a positive real number ε such that for all natural numbers k larger than 1 holds

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

where d(n), the divisor function, denotes the number of divisors of n. The first superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200... (sequence A002201 in the OEIS).

In some sense, a superior highly composite number can be described as having a higher number of divisors scaled relative to the number itself than all other numbers.

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. 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 $p\geq 2^{x}$.

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 $I\subset \mathbb {R} ^{+}$ 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 the OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.

References

• Srinivasa Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. 14, 347-407, 1915; reprinted in Collected Papers (Ed. G. H. Hardy et al.), New York: Chelsea, pp. 78–129, 1962
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300.

External links

• Weisstein, Eric W. "Superior highly composite number". MathWorld.