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Weird Numbers

Definition: A natural number n is an "abundant number" if the sum of its proper divisors is greater than n.

For example, the number 12 is abundant because its set of divisors is {1,2,3,4,6,12} and the sum of its proper divisors is 1 + 2 + 3 + 4 + 6 = 16 > 12.

Definition: A natural number n is a "semiperfect number" if the sum of a subset of its proper divisors is n.

For example, the number 12 is semiperfect because its set of divisors is {1,2,3,4,6,12} and the sum of its proper divisors is 2 + 4 + 6 = 12.

Definition: A "weird number" is a number which is abundant but not semiperfect.

Question: Find all of the weird numbers less than 10,000.

Note: It is now known that there are an infinite number of weird numbers. In the early 1970s, the famous mathematician Paul Erdos offered $10 for the first example of an odd weird number and $25 for the first proof that no odd weird numbers exist. Do you think anyone has capitalized on this bet?

 

Source: Adapted from Problem E 2308, American Mathematical Monthly, September 1972.


Hint: First, determine a list of the abundant numbers. Some patterns will become apparent. For example, any positive multiple of 12 is an abundant number.

Now, try to find an efficient way to check which of these abundant numbers are semiperfect. As a hint, the first weird number is less than 100.

 


Solution Commentary: An effective way to determine the weird numbers is through the use of a computer program (or calculator program).

Whatever your approach, the resultant answers should be: 70, 836, 4030, 5830, 7192, 7912, and 9272.

If you want to know more about the weird numbers, consider the MathWorld website.