# 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 1680

- 1680 is the smallest number with 40 divisors
- 1680 is the smallest area of a triangle with sides, area, and two medians being integers
- 1680 is the smallest number n such that 2n − 1 is less than σ(n) &minus σ(n-2). Equivalently, 1680 is the smallest n such that the sum of non-divisors of n is less than the sum of non-divisors of n-2
- 1680 is the smallest number n such that 2n - φ(x) = σ(x)/2

## Common Properties of 1680

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.

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.