# 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 58

- 58 is the smallest Smith number with a prime sum of digits
- 58 is the number of distinct and commutative semigroups of order 4

## Common Properties of 58

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.

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.