# 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 126

- 126 is the number of prime days in a non-leap year
- The prime gap that covers the first century with no primes (from 1671800 to 1671899) has length 126 (from 1671781 to 1671907)
- 126 is the smallest multidigit number such that the sum of its distinct prime factors equals the product of its digits: the prime factors of 126 are 2, 3, and 7, that sum to 12, which is the product of the digits 1, 2, and 6
- 126 is the smallest vampire number
- 126 is the number of intersections of diagonals inside a regular nonagon
- 126 is the smallest non-palindrome that is a product of 4 primes whose reverse is also a product of 4 primes

## Rare Properties of 126

The number n is called a *vampire* number if there exists a factorization of n using n's digits (counting multiplicity).

## Common Properties of 126

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.

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