# 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 128

- 128 is the largest number which is not the sum of distinct squares
- 128 = 2! + 3! + 5!
- 128 is the smallest number n such that dropping the first and the last digit of n leaves its largest prime factor
- 128 is the least integer greater than 4 which is not the sum of two distinct powers of two and a prime

## Rare Properties of 128

A number is a *power of 2* if it is 2 to some power.

## Common Properties of 128

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

Composite numbers are opposite to prime numbers.

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

Compare with perfect and abundant 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 *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil 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.