# 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 42

- The number 42 is The Ultimate Answer to The Ultimate Question of Life, the Universe and Everything
- 42 is the number of spots on a pair of dice (submitted by Ken Knowlton)
- 42 is the alphanumeric value of FIVE
- The magic constant of a magic 3-by-3-by-3 cube is 42
- Number 42 on the web:
- 42 is the smallest abundant odious number
- 42 is the smallest sphenic Catalan number, where sphenic means a product of three distinct primes

## Rare Properties of 42

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 *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.

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

They are twice triangular numbers.

## Common Properties of 42

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

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

Guess what evil numbers are.

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.