# 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 441

- 441 is the sum of the first 6 cubes
- 441 is the smallest square that can be written as a sum of 3 distinct nonzero squares in 3 or more ways
- 441 is the smallest power that is equal to a sum of first n primes plus 1

## Rare Properties of 441

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

## Common Properties of 441

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.

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

Guess what odious numbers are.

A number is *odd* if it is not divisible by 2.

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

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 next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.