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:

Sources Edit

  1. Odd abundant numbers - OeisWiki

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