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Abundant Numbers

Date: 03/27/99 at 10:47:02
From: Ashley
Subject: Math: perfect, deficient, and abundant numbers for 1-50

I need to find all the perfect, abundant, and deficient numbers 1-50.  
I have already listed all the factors for each number from 1-50 but I 
must be missing some because I have only found a couple of abundant 
numbers. I know there are more

Date: 03/28/99 at 11:21:40
From: Doctor Schwa
Subject: Re: Math: perfect, deficient, and abundant numbers for 1-50

There really aren't all that many abundant numbers. All the multiples 
of 6 (12, 18, 24, 30, etc.) are abundant, because 6 is perfect 
(30 < 1 + 2 + 3 + 5 + 6 + 10 + 15, for example).

Other than that, are there any others? Maybe some things that are 
combinations of 2s and 5s, like 20 or 40?  20 < 1 + 2 + 4 + 5 + 10,
indeed, so it and 40 are abundant.

None of the odd numbers is abundant until you get to pretty big
numbers. I think 945 is the smallest, maybe. So you only need to check
the evens. We already know

  2, 4 are deficient
  6, 28 are perfect
  12, 18, 24, 30, 36, 42, 48 and 20, 40 are abundant.

Let's see, that leaves an awful lot of them,
8, 10, 14, 16, 22, 26, 32, 34, 38, 44, 46, 50

If a number is a power of 2, it is always one short of being perfect,
so it's deficient (example 8: 1+2+4 =7).

If a number is 2 times a prime, it must be deficient (in general, 2p
has factors 1, 2, and p, which add up to less than 2p unless p = 3).

That crosses out most of the remaining numbers (they are deficient)
and leaves us with only a couple more to double-check. I think they 
turn out to be deficient too, so I think there are only those 9 
abundant numbers. But you should check that for yourself.

- Doctor Schwa, The Math Forum
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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