# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 1728

- 1728 is the only compositorial cube (submitted by Sergei Bernstein)

## Rare Properties of 1728

The n-th *compositorial* is the product of the first n composite numbers.

Compositorial numbers are factorials divided by primorials.

- 4,
- 24,
- 192,
**1728**, - 17280,
- 207360,
- 2903040,
- ...

The number n is a *cube* if it is the cube of an integer.

## Common Properties of 1728

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

A number is *even* if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

The number n is *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.

The *untouchable* numbers are those that are not the sum of the proper divisors of any number.