# 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 4

- 4 is the only compositorial square
- 4 is the only positive number that is both the sum and the product of the same two integers
- 4 is the order of the smallest non-cyclic group (submitted by Sam Steingold)
- Every positive integer is the sum of at most 4 squares
- 4 is the smallest number of colors sufficient to color any planar map
- 4 is the only number in the English language for which the number of letters in its name is equal to the number itself
- 4 is the only composite number that is equal to the sum of its prime factors
- 4 is the only composite number n which doesn't divide (n-1)!

## Rare Properties of 4

The n-th *cake* number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.

Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.

The n-th *compositorial* is the product of the first n composite numbers.

Compositorial numbers are factorials divided by primorials.

A k-digit number n is called *narcissistic* if it is equal to the sum of k-th powers of its digits. They are also called *Plus Perfect* numbers.

A number is a *power of 2* if it is 2 to some power.

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

A *tetrahedral* number is the number of balls you can put in a triangular pyramid.

This is the space generalization of triangular and square numbers.

## Common Properties of 4

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.

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.

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 :-) ?

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