An **abundant number** is a number whose proper divisors sum to a value greater than itself. For example, 36 has proper divisors 1, 2, 3, 4, 6, 9, 12, 18, which sum to 55, and \(55 > 36\). Contrast *deficient numbers*, whose proper divisors sum to a smaller value, and *perfect numbers*, whose proper divisors sum to themselves.

The first few abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, ... Note that most of these appear to be even; the first odd one does not appear until 945. This is quite remarkable, as it provides a naturally occurring example of a large number. After 945, the odd-abundant numbers are 1575, 2205, 2835, 3465, ...

There are also abundant numbers whose proper divisors have a sum greater than *twice* the original number. The smallest one is 180, but no odd ones occur until 1018976683725.

## Notable examples Edit

These are special odd-abundant numbers, such as:

- 5391411025 is the first odd-abundant number that is not a multiple of 3.
- 20821017304425168561312837502762890375 is the smallest odd number whose proper divisors have a sum greater than
*three times*the original number. - 48870871124826570463953805139878697155358000962012333290725030523875 is the smallest odd number that is not a multiple of 3
*and*whose proper divisors have a sum greater than*twice*the original number. - 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701 is the smallest abundant number that is not a multiple of primes less than 13.
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