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June 19, 2016 Comments (3) Views: 1329 Mathematics

Defining “Weird Numbers”

Weird. A word that has stuck with us for many centuries, growing first from the Old English wyrd, as in fate. Decades pass, suddenly Shakespeare uses it to describe a trio of witches in Macbeth, and henceforth it is grouped along with occult terms like invocation, lucidity, wraith. Sometime after this, it loses that religious connotation, but its basic grouping remains: not of the norm.

Fast forward to now. A student works through a problem their teacher has assigned. After minutes of work, the students comes to an answer: 76/21. The student nervously eyes the number and thinks, “What a weird number.”

As it turns out, mathematicians have actually defined what a weird number is, and, well, it’s probably not what you expected it to be. As Wikipedia says, “In number theory, a weird number is a natural number that is abundant but not semiperfect.” But what does that mean? Let’s take a look at a few definitions that will help us understand this:

First off, what’s number theory? It’s a subsection of pure mathematics that deals with, well, numbers. Specifically, it’s concerned with things like prime numbers (How many are there? How often do they occur?).

What are the natural numbers? They’re basically what we tend to think of as “counting numbers”, e.g. 1, 2, 64, 100, 120000. These numbers are never negative. There’s some discussion about whether or not we include 0 in this group—it really depends on who you ask, but personally I’ve always been taught not to include it. Pick your poison, it won’t change our discussion here.

What does abundance mean? This is where it starts to get tricky. Wikipedia tells us that an abundant number is one “for which the sum of its proper divisors is greater than the number itself.” (Quick reminder: the proper divisors of number A are all of the numbers B, C, D, and so on that multiply together to get number A.) To make this clear, let’s look at an example: 12.

12’s proper divisors are: 1, 2, 3, 4, and 6. We add those up and get 16, which is 4 more than 12!

weird numbers 12 is abundant

Contrast that with a number like 14. Its proper divisors are 1, 2, and 7—which sum to 10. We call numbers like this, where the proper divisors add up to less than the number itself, deficient numbers.

Now that we’ve cleared that up, we can define semiperfect: any natural number that is equal to the sum of all or some of its proper divisors. Once again, let’s look at 12:

Among the set of 12’s proper divisors, consider: 2, 4, and 6. Adding those up, we get 12!

weird numbers 12 is semiperfect

Now then, with all of our definitions arranged, we can finally look back and consider what a weird number really is. A reminder of that definition: “a natural number that is abundant but not semiperfect.” Let’s consider this by example:

15: Divisors are 1, 3, and 5. 1+3+5=9, which is less than 15. Thus, 15 is deficient, and not a weird number.

36: Divisors are 1, 2, 3, 4, 6, 9, 12, and 18. Summing these gets us to 55, which is bigger than 36, making this abundant. However, 6+12+18=36, which makes this semiperfect. Thus, not weird.

70: Divisors are 1, 2, 5, 7, 10, 14, and 35. Add these up, and you get 74, making this abundant. However, no combination of these divisors gets us to 70. We have our first weird number! In fact, 70 is actually the first weird number. The next smallest weird number is 836. Don’t believe me? Feel free to take a look for yourself, now that you know what a weird number looks like.

Number theory has a lot of fun concepts like this. If you enjoy numbers for numbers sake, this is one of the best places to start looking into mathematics, as many of the basic (but still interesting) concepts only require addition and multiplication.


Interesting Note: Weird numbers are actually the topic of an unsolved question in mathematics. As of now, we don’t know if there are any odd weird numbers. Unlike some of the other big unsolved math questions, there is no monetary reward for finding this out.

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3 Responses to Defining “Weird Numbers”

  1. JimP says:

    Quick question for you: Why is 1 considered a divisor of a number but the original number itself does not make your list? I realize that if the original were used, then it would be impossible to have a deficient number. However, don’t we typically only consider a number to be a divisor if it can be multiplied by another number to reach the original? For B to be a divisor of a number A, must B < A but not equal? Is the original number ruled out when talking about PROPER divisors? Just wondering.

    • Wyatt Ehlke says:

      Yes, the original number is ruled out when we talk about proper divisors. We still include 1 because 1 is a divisor of every number, and our set of proper divisors also includes every divisor of THOSE numbers. For example: One of 12’s proper divisors is 6. The proper divisors of 6 are 1, 2, and 3, which are also proper divisors of 12.

      That’s why the proper divisors of 4 are 1 and 2. 2 times 2 is 4, which means that 2 is a divisor. And 2 times 1 is 2, which means that 2 and 1 are divisors of 2, which means they’re included in the set called “proper divisors of 4.”

      Does this lend some clarity? If not, let me know and I’ll try to type up a clearer example.

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