Abundant Numbers Day - December 18

Happy Abundant Numbers Day!

What are abundant numbers (aka excessive numbers)? They are related to the concept of perfect numbers, which are the numbers where the sum of the proper divisors is equal to 2 times the number itself. Abundant numbers are where the sum of the proper divisors is greater than the number. (The remainder, numbers where the sum of the proper divisors is less than the number, are called deficient numbers.)

Here is the start of the set of abundant numbers.


Every natural number is either perfect, abundant, or deficient. Here's the arithmetic that determines some of the early ones.

We celebrate Abundant Numbers Day today because 12 and 18 are the 2 largest consecutive abundant numbers that form a date: 12/18 or December 18.


Comments

  1. andyboyDecember 18, 2022 at 11:10 PM

    Why isn't 2 an abundant number? 2 + 1 = 3 which is greater than 2.

    ReplyDelete
    Replies
    1. jpvioletteDecember 19, 2022 at 10:47 AM

      Good pickup. My description was slightly wrong (which I've corrected). Basically the definition of abundant numbers requires the sum of "proper divisors" to exceed the number itself. So in your example, 2 isn't included in the sum because 2 isn't a proper divisor, leaving the (trivial) sum as just plain old 1, which is less than 2, and putting it in the deficient number category.

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