# 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 199

- 199 is the smallest number with an additive persistence of 3
- 199 is the smallest prime which can be partitioned into three parts such that the last part is the product of the first two
- 199 is the smallest non-cubic number whose fifth power is closer to a cubic number than a square number
- 199 is the smallest multidigit prime such that its largest digit is equal to the largest digit of the sum of its digits

## Common Properties of 199

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 *odd* if it is not divisible by 2.

Numbers that are not odd are even. 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.

A *prime* is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

A number is said to be *square-free* if its prime decomposition contains no repeated factors.

A prime number is called a *twin* prime if there exists another prime number differing from it by 2.