Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/