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.