# 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 130

- 130 is the smallest number n such that the n-th Fibonacci number contains a pandigital substring
- 130 is the smallest out of a mutually-praising pair of numbers
- 130 is the only integer that is the sum of the squares of its first four divisors: 130 = 1
^{2} + 2^{2} + 5^{2} + 10^{2}

## Rare Properties of 130

The n-th *cake* number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.

Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.

## Common Properties of 130

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

Guess what odious numbers are.

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.

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