# 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 210

- 210 is the smallest area of two different primitive Pythagorean triangles (20, 21, 29 and 12, 35, 37)
- 210 is the product of the first 4 primes
- 210 is the smallest number such that the sum of the 4th powers powers of its digits equals the sum of its prime divisors: 210 = 2*3*5*7 and 2
^{4} + 1^{4} + 0^{4} = 2 + 3 + 5 + 7 = 17

## Rare Properties of 210

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 210

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

Guess what odious 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.

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 *untouchable* numbers are those that are not the sum of the proper divisors of any number.