# 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 45

- 45 is the only number that is the sum of its digits multiplied by 5
- 45 is the smallest deficient number that is not distended, where distended numbers are positive integers n for which each divisor of n is greater than the sum of all smaller divisors
- 45 is the smallest composite super-Catalan number
- 45 can be partitioned into two parts that concatenate to its square: 45 = 20 + 25 and 45
^{2} = 2025. Also, 45 can be partitioned into three parts that concatenate to its cube: 45 = 9 + 11 + 25 and 45^{3} = 91125. Moreover, 45 can be partitioned into four parts that concatenate to its fourth power: 45 = 4 + 10 + 06 + 25 and 45^{4} = 4100625

## Common Properties of 45

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.

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.