divisors

[miriam_e]

Here is something weird. I ran a quick little program today to see what numbers up to 1,000 are most divisible by other numbers. After collecting the results and running them through a sort I became a little puzzled at how the number of divisors seems to collect around certain particular amounts, so I charted the unsorted results to take advantage of our wonderful visual pattern perception. What I saw was a bit of a surprise. It looks really familiar, and I'm not quite sure why.

The horizontal axis lists the numbers between 1 and 1,000 examined.
The vertical axis how many other numbers the number could be divided by. The number itself and 1 were not included, so prime numbers get listed as having zero divisors.

The really interesting numbers here are the ones on the top of the curve, which are (going from top right down to bottom left):
840, 720, 360, 240, 180, 120, 90, 48, 36, 24, 12, and 6
Next time you wonder if the Babylonians had rocks in their heads sticking us with such crazy numbers for calculating angles, think again. They are brilliant numbers. They divide up more easily than any others.

There are some really noteworthy things.
Primes are pretty uncommon, you'd think, but there are 168 of them. Far more uncommon are almost-primes -- numbers that have only one other divisor. There are only 11 of them in the first 1,000 numbers (4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961).
Only 3 numbers have just 3 divisors (16, 81, 625).
Only 2 numbers have 5 divisors (64 and 729).
There are no numbers below 1,000 with 9 or 11 or 15 divisors, with many more higher up (the gaps become more frequent as you look for greater numbers of divisors).

I wonder why the number of divisors falls into such prominent bands on 0 (prime), 2, 4, 6, 10, 14, and 22. Especially 14! That is such an odd number. It is nothing like the almost random distribution I'd have expected.

Comments

[cranky--crocus.livejournal.com]

That is so fascinating. Just...whoa. I really liked reading all that. I should give it to my maths teacher. My classmates are always wondering why the Babylonians "randomly" picked angle numbers and such things.

[miriam-e.livejournal.com]

Heheheh :)
I love the reason the Babylonians made 360 degrees in a circle. It was a mistake. :) They calculated 360 days in the year and thought they'd stumbled on one of the universe's great secrets because 360 is such an utterly wonderful number. It can be cleanly divided by 22 other numbers:
2 3 4 5 6 8 9 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180

But in the end the joke was on them. The year is really 365.25 days long, which is just a random, ugly number. :)

[cranky--crocus.livejournal.com]

Snork. Leap year!

While I was working for a medical billing company inputting patients I finally found someone born on a leap day. I had never seen it before. Grin. I was so proud.

[miriam-e.livejournal.com]

heheheh :)

On the plus side they don't have to lie about their age. They are always a quarter of the birthdates of other people. :)

[cranky--crocus.livejournal.com]

Mmm...it doesn't start sounding good until they're at least 72, though, otherwise everyone they're involved with seems like a pedophile. xP.

[annie-lyne.livejournal.com]

It is easy to see that n! will have lots of divisors, which are some of the numbers which you quote above.

[miriam-e.livejournal.com]

The factorials, n! under 1,000 are (if my crappy maths serves me) 1, 1, 2, 6, 24, 120, 720. Yes. It does give an intuitive feel for why they figure so high on the list. And it may even have something to do with some of the other numbers (6, 12, 24, 36, 48, 90, 120, 180, 240, 360, 720, 840) as being products of smaller factorials.

The main reason I did this is that I'm so mathematically inept I wanted to actually see the the numbers that have the most divisors.

What really struck me was something I wasn't looking for: the strange way the amount of divisors cluster, with so many having 0, 2, 4, 6, 10, 14, or 22 divisors, with almost none in between. (Or, if you count the number itself and 1 the values are 2, 4, 6, 8, 12, 16, 24... which looks rather misleadingly normal.)

Nevertheless, there are 97 numbers that have 10 divisors, but none that have 9 divisors or 11 divisors. That strikes me as odd. Why 10? (12 if you count the number itself and 1.)

There are 22 that have 8 divisors, but that is still way less than 97, and there are only 8 that have 7 divisors. There are only 5 that have 12 divisors, and only 4 that have 13.

Why does 10 stand out so strongly?

Likewise with the other prominent bands. I'd have expected that there would be a more randomly graded pattern concentrating more to the middle of the range. Instead we get these weird bands that look a little like quantum energy levels.

[annie-lyne.livejournal.com]

Interesting questions. I will play around and see if I can cook up anything.

Note that a "divisor" counts the number itself and 1, whilst a "proper divisor" excludes them. It is more natural to talk about divisors rather than proper divisors, but this is just a difference of 2.

[annie-lyne.livejournal.com]

Any number of the form p^k will have k+1 divisors, where p is prime. The smallest number which has 11 divisors then is 2^10 = 1024.

There can be more work done to show that any number of the form p^kq^j can never have 11 divisors, etc.

[miriam-e.livejournal.com]

Cool! That is interesting. Great food for thought. Thank you.

I've just realised I made a small oversight. In my list I only considered numbers that have simple factors of just 2 numbers multiplied together. I didn't consider those that result from 3 numbers multiplied together (for instance 12 can result from 2*6 or from 3*4, but it can also result from 2*3*2) or greater amounts of numbers multiplied together. The list is still interesting though because of the number of things that come naturally out of such simple two-part operations (for example much of cryptography involves deliberately restricting stuff to just 2 parties).

I'm starting to understand the patterns now. There may be some simple mathematical way of looking at it, but I see it from a procedural standpoint. The pattern grows because of the way some numbers are more divisible than others, and why they are.

For instance:

The numbers that have 0 divisors -- the primes -- there are a lot more of them than I'd expected, but they are like the foundation for everything else.

The numbers with 1 divisor are squares of primes. It doesn't take long for them to grow beyond 1000 -- 31, the 11th prime (not counting 1) squared is 961 already, so it's easy to see why there would be very few of them (only 11 in the positive integers less than 1,000).

The numbers with 2 divisors -- there are lots of them (292) -- are the product of two primes. There are lots of primes so you'd expect lots of numbers with 2 divisors. I haven't looked at them properly yet.

The numbers with 3 divisors are weird. There are only 3 of them (16, 81, 625) and their factors are a prime, the square of that prime, and the cube of that prime. This is because they are the fourth power of a prime (2, 3, 5). It is easy to see why there would be so few of them. The multiplication n*n*n*n can be broken down just 2 ways: (n*n)*(n*n) and n*(n*n*n). There you have your 3 factors.

There are 110 numbers with 4 divisors. I'd expect there to be lots, but I haven't really looked at them yet.

More interesting is why there are only 2 with 5 divisors: 64 and 729. It is easier to see why when you look at them. 64: 2 4 8 16 32 and 729: 3 9 27 81 243 They are progressions of powers like the numbers with 3 divisors and they break apart exactly the same way. They are the 6th power of a prime. n*n*n*n*n*n They can be broken apart as simple multiples of two numbers only 3 ways to give 5 different numbers: n*(n*n*n*n*n), (n*n)*(n*n*n*n), (n*n*n)*(n*n*n). What makes this arrangement so uncommon is that having an even power means that one of the pairings must be a square, and squares grow out of control really quickly.

and so on...

It is like a messy, but interesting, sieve. The way the groups of numbers dance around and pair up is quite pretty, but at times a bit overwhelming.

Oh god I waste time, don't I?
I'm supposed to be working on a picture and looking into whether I can do some VR work for a friend in Western Australia.
Gah!!!

[annie-lyne.livejournal.com]

Here is a trick if you want to work out why certain numbers have so few divisors. Each number can be written as 2^a 3^b 5^c ... p^k, and so on -- the prime decompositions. The number of divisors can be found by multiplying each exponent plus 1 together.

For example: 2^2*3*5^3 = 1500. 1500 has (2+1)(1+1)(3+1) divisors.

[dorjejaguar.livejournal.com]

It's looks to me like grass growing. Or perhaps more accurately grass advancing/colonizing. Or perhaps like the edge of a forest, the way it reaches out into the non-forest.

[miriam-e.livejournal.com]

It does, doesn't it.

Perhaps it can be used as a simple rule to simulate such things in the virtual worlds I build. I doubt that it actually is the rule-set underlying them, but it could still be useful... and it is rare that we stumble upon things like the 3 simple rules that Craig Reynolds discovered governing seemingly complex mass movements such as how birds flock, beasts herd, and fish school.

Thanks for the nice alteration in my viewpoint jaguarnoelle.

[dorjejaguar.livejournal.com]

You're welcome. :)

It's obvious to me that your brain knows about things my brain does not. Which is lovely. I do think I'm gonna have to research some of these things you're talking about to be able to talk with you about them though. Thanks for giving me things to think about. :)