# 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 22

- 22 is the smallest Hoax Number
- 22 is the smallest multidigit number such that the sum of its digits equals the product of its digits
- 22 is the only fixed point of the Look and Say operation
- 22 is the smallest number that can be partitioned into sets integers such that the sum of their reciprocals is 1 in more than one way
- 22 is famous as the numerator of of 22/7, commonly used rational approximation for pi with small denominator
- When cutting a circle with just six line segments, the maximum number of pieces that can be so created is 22. In addition to that, cutting a circle with five circles, the maximum number of pieces is also 22
- 22 is a catch, as in Catch-22
- 22 is the only palindrome with an even number of digits that can be expressed as a sum of two palindromic primes
- 22 is the smallest number that is the sum of terms of an arithmetic progression containing four positive different integers in two ways
- 22 is the smallest integer n, except for the trivial 1, such that n! has n digits
- There are 22 pentahexes

## Common Properties of 22

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 n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

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

Guess what evil numbers are.

A *palindrome* is a number that reads the same forward or backward.

A composite number is called a *Smith* number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?

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