Types of Numbers

If anyone knows what an untouchable number is, we will give you $10 right now (rounded to the nearest $100). No one? Yeah, that’s understandable. Mathematicians have invented names for way too many types of numbers. To prove our point, let’s run through seven different examples of extremely pointless types of numbers.

Let’s start with a fun type of number: Vampire numbers. Vampire numbers are numbers whose digits you can rearrange and separate into two smaller numbers of equal length (the fangs) that multiply to the original number. The smallest such number is 1,260, which you can split into 21 and 60, which multiply to 1260, the original number. Some other examples are 1,395, 1,435, 1,530, and 1,827. Ok, now let's look at the uses of vampire numbers. Hmm…. Oh, wait. There are none. Vampire numbers are completely useless in mathematics. How pointless.

Let’s see another example: untouchable numbers (what an amazing name). Untouchable numbers are numbers that cannot be expressed as the sum of the proper divisors of some positive integer (the “aliquot sum” of the number). So, a number like 4 is not untouchable as it can be represented as the sum of the proper divisors of 9: 3 + 1 = 4. Examples of untouchable numbers are 2, 5, 52, and 88. See if you can convince yourself that these numbers cannot be… what is the word? Touched? Of course, this is utterly useless.

Related to untouchable numbers are amicable and sociable numbers. Amicable numbers are two numbers that are each equal to the sum of the proper divisors of the other (aka aliquot sum). The most famous of these are 220 and 284: 220’s proper factors are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, summing to 284, and 284’s proper factors are 1, 2, 4, 71, and 142, summing to 220. Often, amicable numbers represent friendship or love, with each person having a keyring, amulet, or something similar with an amicable number engraved on it. 

Sociable numbers are very similar - they are three or more numbers that form a loop when you continuously take the aliquot sum. Weirdly, no such sequence of exactly three numbers has been discovered, but we’ve discovered over 5,000 with length 4. There are only a few of length 5, 6, 8, and 9, and then none until 28! Despite being incredibly fun to play with, they also have no practical use: as the famous mathematician John Conway said, “The only application or use for these numbers is the original one - you insert a pair of amicable numbers into a pair of amulets, of which you wear one yourself and give the other to your beloved!”

One other interesting (or not) species of numbers is the odious numbers. These are numbers that, when represented in base 2 (binary), have an odd number of 1s. So, the first odious numbers are 1, 2, 4, 7, and 8, which in binary are written as 1, 10, 100, 111, and 1000. Numbers that are not odious have an equally fun name: evil numbers! These are numbers that have an even number of 1s in binary. According to Wikipedia, the only use of these numbers is that “the partition of evil and odious numbers provides a solution to the Prouhet-Tarry-Escott Problem,” which itself is useless, as well as having many simpler solutions. (And for that reason, we won’t dive into the meaning of “partition” here.)

There are also narcissistic numbers, which are named because they always seem obsessed with the number of digits they have: they are numbers equal to the sum of the nth powers of their digits. For example, 153 is narcissistic because 153 = 13 + 53 + 33. Fun fact: they are also called Armstrong numbers. Not the Armstrong guy on the moon (though he might also be narcissistic), but Michael F. Armstrong, an American lawyer. As you might have guessed, these numbers are completely irrelevant to the rest of mathematics. As the distinguished mathematician G. H. Hardy wrote, “there is nothing in [narcissistic numbers] that appeals to the mathematician.” 

For another example, we have a type of prime: the “emirp.” Before you read on, can you guess what property they have? Hint: the answer’s in the name. These are primes which are still primes when written backwards - thus the name, which is simply “prime” reversed. These are fairly common, with 36 emirps under 1,000. One instance of an emirp is 37, as it and its reverse, 73, are both prime. Since reversing a number has no mathematical significance, the fact that a number is an emirp is, yet again, very much meaningless.

We’ve saved the most silly, stupid, and complicated gibberish for last. They are the Superior Highly Composite Numbers. Here is the first line on the Wikipedia page for this type of number: “In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors.” What an overcomplicated name for such a simple idea. Some examples of this type of number are 2, 6, 12, and 60. Except that’s not the full definition. Formally, it is defined as numbers such that for:

(here, d(n) means the number of factors that n has). If that sounds like a big random mess of math notation, we quite agree. Technically, these numbers are often used as number bases (base 2, 6, etc.), but aside from that, these numbers are completely pointless.

Congrats! You’ve learned quite a bit today! You’ve learned seven different types of numbers! Of course, unlike conventional knowledge, we can almost guarantee that you will never, ever use this knowledge in your life. So, it should be clear that we have way too many types of numbers–or too many mathematicians. If you need more convincing, feel free to search up Munchausen numbers, Pentatope numbers, and Weird numbers. Have fun!