# 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 70

- 70 is the smallest weird number
- 70 is the largest known number n such that 2
^{n} has a digit sum of n (the only other such number known is 5)
- 70 is the only non-trivial square number of cannonballs that can be piled into a square pyramid (submitted by Ken Knowlton)
- 70 is the smallest pentagonal number which is a product of 3 distinct primes (submitted by Massimo Kofler)
- 70 is the smallest number n such that the sum of proper divisors of n exceeds n, yet no subset of them adds to n

## Rare Properties of 70

The number n is called a *weird* number if it is abundant but is not the sum of any subset of its proper factors.

## Common Properties of 70

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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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.