# 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 540

- 540 is the smallest number whose divisors contain every digit at least twice
- 540 is the smallest positive heptagonal number that is also a half of a pentagonal number
- 540 is the smallest number n that is a concatenation of two other numbers s and t such that n is the product of the sums of divisors of s and t (540 = 6*90 = σ(5)*σ(40))

## Common Properties of 540

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.

The number n is called an *apocalyptic power* if 2^{n} contains the consecutive digits 666 (in decimal).

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.

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.