#triangle numbers

So I was thinking a little bit about sums, and alternate product formulas for writing out sums.  The formula for triangle numbers (aka the sum of the first n natural numbers) of course being perhaps one of the most famous formulae of all time:

The first few triangle numbers are 1, 3, 6, 10, 15, 21, 28, 36... you might be wondering why I’m bothering to list them, but I’ll be getting to that shortly.

Another formula I was thinking about was the formula for computing perfect numbers.  To recap: a perfect numbers is a numbers whose factors (other than itself) add up to itself.  E.g. 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14.  There aren’t many perfect numbers, the list starts out 6, 28, 496, 8128 before leaping up to 33,550,336 and then just kinda going crazy from there.

Now the formula for even* perfect numbers is another famous mathsy formula, and is as follows:

where 2^p - 1 is prime (such prime numbers being called Mersenne primes).  3 (4-1) is prime, and so 6 (2x3) ends up being perfect.  7 (8-1) is prime, so 28 (4x7) ends up being perfect.  Then we skip up to 31 being prime, which leads to 496 (16x31) being perfect, and 127 (128-1) being prime, which leads to 8128 (64x127) being perfect.  Followed by a big ol’ leap to 8191 (8192-1) being prime and 33,550,336 (4096x8191) being perfect.

I’m honestly not sure if I’m spending too much or too little time explaining this stuff but at this point I really just want to get to the galaxy brain moment, which is:

or in other words “hey, the triangle number formula and the perfect number formula look weirdly similar.  what if... what if I just take a factor of ½ out of the perfect number formula?”

A little more arranging gives us:

So every even* perfect number is the (corresponding Mersenne prime)th triangle number!  And if you look back at the triangle numbers I listed above, sure enough, there are 6 and 28 bold as brass as the 3rd and 7th triangle numbers.  496 is the 31st triangle number, 8128 is the 127th triangle number, 33,550,336 is the 8191st triangle number, and so on and so forth.

All right.  After all that, back to the actual dang puzzle.  Well, let’s go back to our original definitions of what triangle numbers (adding up consecutive natural numbers) and perfect numbers (adding up factors of a number) involve!  For 6 we get the fascinatingly trivial

1 + 2 + 3 = 1 + 2 + 3

Cancelling down, this just gives us our first term, 0 = 0.

For 28 it gets more fun and was the starting point idea for this puzzle:

1 + 2 + 3 + 4 + 5 + 6 + 7 = 1 + 2 + 4 + 7 + 14

which cancelled down gives us our second term

3 + 5 + 6 = 14

It’s odd to think that this simple sum can be hiding so much behind it!

The third term is the same logic applied to 496, on the LHS we have the sum of the natural numbers from 1 to 31 and on the other side we have the sum of the factors of 496 (other than itself).  And hopefully you’re starting to see why doing this for the fourth term felt nightmarish, and the fifth term was unthinkable!

In some ways this is a kind of barmy puzzle but hey, I had an idea and investigated where it went and something vaguely puzzle-like popped out.  That’s how most of the puzzles on this blog work anyway.  See you in another year or something, I guess.

While seemingly logical, this was a no-go. Ever slow, perhaps it would have worked if I let it just run on my machine for the night, but instead I turned to my pal Histocrat, always there to lend a hand.

The difference here is that instead of running through all of the numbers, we instead find all of the divisors for the square root and then multiply them by 2. This is not entirely intuitive for me, but is apparently true. In any case, the code ran significantly faster and we were able to get a workable answer. 

From there, Histocrat decided to teach me a lesson in classes:

I had learned about classes before, but to be honest I've been sort of taking a break from them and I think it was a good call. Now that I have been coding with only methods, the use for a class has become clear (or clearer anyway). Codecademy likes to talk about Classes with Persons and attributes of that person and such or classes of Animals with specific creatures defined beneath. While the semantics appear understandable through this process and a hierarchy is clear to piece a part, I find the purpose of such items are a little fuzzier until you start to go a little deeper. At least that was my experience. 

Using this TriangleNumber class we basically have this lovely container filled with an infinite amount of TriangleNumbers (sort of like our Universe Man, above, no?) Here we can easily access and do things to it. Pretty sweet.