Abundant numbers are numbers whose factors add up to a number larger than the original number (see also 12). The sequence starts 12, 18, 20, 24, 30, 36, 40, 42, 48... note that those abundant numbers are all even.

Therefore, you may think that there are all even, or perhaps wonder if there are any odd abundant numbers at all.

There are indeed odd abundant numbers; however the smallest isn't until 945 - with a sum of factors = 975.

List three more odd-abundant numbers.

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

**1,018,976,683,725.
**