# 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 6

- 6 is the only even evil perfect number (submitted by Alexey Radul)
- 6 is the order of the smallest non-abelian group (submitted by Sam Steingold)
- 6 is the only number (except 1) such that the sum of all the primes up to 6 equals the sum of all the composite numbers up to 6 (inclusive)
- 6 is the only even perfect number, for which repeatedly summing the digits you do not get 1
- 6 is the only mean between a pair of twin primes which is triangular
- The symmetric group S
_{6} is the only finite symmetric group which has an outer automorphism
- 6 is the minimum number of colors that is always sufficient to color any map on a Klein bottle or on a Möbius strip
- 6 is the smallest perfect number
- 6 is the only number that is both the sum and the product of the same three distinct positive integers
- 6 is the only square-free perfect number (submitted by Alexey Radul)
- A web page about 6: Some Thoughts on the Number Six, John Baez
- 6 is the only even perfect pronic number
- 6 is the largest integer to be both a factorial and a primorial
- 6 is the only perfect number that is also a product of its proper divisors (submitted by Michael W. Ecker)
- 6 is the largest square-free factorial
- 6 is the only perfect factorial (submitted by Alexey Radul)
- 6 is the number of convex regular polychora in 4D space (submitted by Carlo Séquin)
- 6 is the largest triangular number whose square is also triangular
- A tetrahedron has 6 edges, a cube has 6 faces, an octahedron has 6 vertices
- The maximum number of circles of same size that fit tangentially around a given circle is 6 (submitted by Chuck DeCarlucci)
- The maximum number of side of a regular polygon that can tile a plane is 6 (submitted by Chuck DeCarlucci)

## Rare Properties of 6

The number n is called an *automorphic* number if (the decimal expansion of) n^{2} ends with n. These numbers are also called *curious*.

It is curious, how for a k-digit automorphic number n there is another automorphic number -- 10^{k} + 1 - n. For this to work with n=1, you have to treat 1 as a zero-digit number.

The n-th *factorial* is the product of the first n natural numbers.

The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.

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.

The number n is *perfect* if the sum of all its positive divisors except itself is equal to n.

Less than perfect numbers are called deficient, too perfect -- abundant.

The p-*primorial* is the product of all primes less than or equal to p. It is sometimes denoted by p#.

Compare to compositorials and factorials.

The number is called *pronic* if it is the product of two consecutive numbers.

They are twice triangular numbers.

## Common Properties of 6

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 *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

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

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.

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

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.

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