# 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 14

- 14 is the largest number for which there are as many composite numbers less than it as there are primes
- 14 is the lowest even n for which the equation φ(x) = n has no solution, making it the first even nontotient
- 14 is the smallest positive integer n such that n and 2
^{n} end with the same digit
- 14 is the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space (submitted by Corey Cheever)
- 14 is the smallest number n such that σ(n) = σ(n+1)
- 14 is the smallest number such that it and the next number are both semiprimes (14 = 2*7 and 15 = 3*5)

## Rare Properties of 14

The n-th *Catalan* number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).

There are many ways Catalan numbers can be interpreted; there are some cool pictures here and the Wikipedia article is very good.

## Common Properties of 14

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

Guess what evil numbers are.

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