# Number Gossip

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

## Rare Properties of 8128

The number n is *perfect* if the sum of all its positive divisors except itself is equal to n.

Less than perfect numbers are called deficient, too perfect -- abundant.

- 6,
- 28,
- 496,
**8128**, - 33550336,
- 8589869056,
- 137438691328,
- ...

## Common Properties of 8128

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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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

Guess what evil numbers are.

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

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.

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