# 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 1296

- 1296 is the total number of rectangles one may draw on a standard chessboard
- 1296 is the smallest number with 25 divisors
- 1296 is the smallest square which when subtracted from its reverse forms a positive square
- 1296 is the smallest semiprime to the power of another semiprime (1296 = 9
^{4})

## Rare Properties of 1296

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

## Common Properties of 1296

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 *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil 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.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.

The *untouchable* numbers are those that are not the sum of the proper divisors of any number.