# 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 2821

The composite integer n is a *Carmichael* number if b^{n-1} = 1 (mod n) for every integer b
which is relatively prime with n.

Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.

## Common Properties of 2821

To build the *lucky* number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, ... . The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, ... . The next number remaining is 9, so delete every ninth number, etc.

Those numbers were lucky they weren't crossed out.