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

[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. :)