Abundant number

An abundant number is a number which divisors add up to more than twice the number itself.

The difference between the added-up divisors and twice the number is known as the abundance.

Example:

132: 132 is an abundant number.

132 has these factors: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132 &mdash; they add up to 326. Twice of 132 makes 264. The abundance is 326 - 264 = 62.

Properties
Multiples of all abundant numbers are always abundant numbers. In the same way, multiples of all perfect numbers are always abundant numbers.

Types of numbers that has abundant numbers

 * Some pronic numbers, except 0, 2 and 6, which are undefined, prime (and therefore deficient), and perfect respectively.
 * Some even and odd numbers, perfect powers. 945 is the first odd abundant number. (These are trivial).
 * Composite numbers. No prime number is an abundant number.
 * Semiperfect number, like 12.
 * Weird number. It was created that it is always an abundant number. 70 is the first weird number.
 * Highly composite numbers. The last 2 or 3 divisors can already prove it is an abundant number.
 * Quasiperfect number. None has been found, and should be an odd square number greater than 1035 and has 7 distinct prime factors.