# 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 36

- 36 is the smallest number out of two (the other being 360) that have the same number of letters in its Roman representation as its double, triple, quadruple, quintuple, sextuple and septuple
- 36 is the smallest number (besides 1) which is both square and triangular
- 36 is the smallest square that is the sum of a twin prime pair: 17 and 19
- On the piano, 36 is the number of black keys
- 36 is the smallest number containing all the digits when raised to the 10th power
- 36 is the smallest number n such that n and 2
^{n} end with the same two digits
- 36 is the smallest abundant square number
- 36 is the smallest abundant triangular number
- 36 is the smallest integer which can be expressed as the sum of consecutive primes in 2 ways: 5 + 7 + 11 + 13 and 17 + 19

## Rare Properties of 36

The number n is a *square* if it is the square of an integer.

## Common Properties of 36

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

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.

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

Guess what odious numbers are.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

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.