Superior highly composite number

All superior highly composite numbers are highly composite. This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e. ${\displaystyle d($k)=d(n)}  , but ${\displaystyle k<n}$ ), then ${\displaystyle {\frac {d($k)}{k^{\varepsilon }}}>{\frac {d(n)}{n^{\varepsilon }}}}   for all positive ε, so if a number "n" is not highly composite, it cannot be superior 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.^[2] Let $${\displaystyle e_{p}($$x)=\left\lfloor {\frac {1}{{\sqrt[{x}]{p}}-1}}\right\rfloor }   for any prime number p and positive real x. Then $${\displaystyle s($$x)=\prod _{p\in \mathbb {P} }p^{e_{p}(x)}}   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 ${\displaystyle s'}$  exists a half-open interval ${\displaystyle I\subset \mathbb {R} ^{+}}$  such that ${\displaystyle \forall x\in I:s($x)=s'}  .

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 the OEIS). In other words, the quotient of two successive superior highly composite numbers is a prime number.

The first few superior highly composite numbers have often been used as radices, due to their high divisibility for their size. For example:

• Binary (base 2)
• Senary (base 6)
• Duodecimal (base 12)
• Sexagesimal (base 60)

Bigger SHCNs can be used in other ways. 120 appears as the long hundred, while 360 appears as the number of degrees in a circle.

• Ramanujan, S. (1915). "Highly composite numbers". Proc. London Math. Soc. Series 2. 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01. 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.

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