Find all the abundant numbers by starting at 1 and checking the proper divisors of each number you consider to see if the sum of the proper divisors add up to more than the number. The smallest such number is 12. The proper divisors of 12 -- the numbers less than 12 that divide into 12 without leaving a remainder -- are 1, 2, 3, 4 and 6. The sum of 1, 2, 3, 4 and 6 is 16, and 16 is greater than 12 so 12 is an abundant number.
Generate an infinite number of abundant numbers by multiplying small abundant numbers by any integer. For example, because 12 is abundant, so is 24, 36, 48, 60 and so on. The multiples of perfect numbers are also abundant. The smallest perfect number is 6. The proper divisors of 6 are 1, 2 and 3 and 1 + 2 + 3 = 6. Multiples of 6 like 12, 18, 24, 30 and so on are all abundant numbers. For example, the proper divisors of 30 are 1, 2, 3, 5, 10 and 15, and 1 + 2 + 3 + 5 + 10 + 15 = 36 so 30 is an abundant number.
Know some of the facts about abundant numbers to help you with the search. The smallest abundant number number is 12. It is also interesting to know that the smallest perfect number is 6 -- so every multiple of 6 is an abundant number. The next perfect number is 28 -- after that perfect become fewer and further between. There are 21 abundant numbers below 100 -- all of them even numbers. The smallest odd abundant number is 945.