User blog comment:Plain'N'Simple/Using triangular numbers to create a Conway-Arrow Level Notation (probably not what you'd expect)/@comment-35434595-20191025155632/@comment-39541634-20191026163729

@Planetn9ne

Yes, tetrahedral numbers would be a natural extension up to w^3. A similar extension can easily be made up to w^a for an arbitrary a.

Generalizing this approach to w^w seems more tricky. At least if you want an easy way to calculate the transformation.

You can easily define an w^w ordering by, say, defining X(v)=[the number of dimensions + the sum of all coordinates] for any vector v, and then writing down all the vectors in order of increasing X. But as far as I know, there's no easy way to calculate the position of a given number in the grid, in this case.

@P-bot

I wasn't aware this was a standard function in the mathematical repertoire. My inspiration was the famous 1-1 correspondence of the rationals to the integers (the proof that Q is countable), and I developed the triangular-based formula in my own. I was really surprised to find how simple the final formula turned out.

Also, I am aware of the prime factorization tactic, but I wanted to do something different here. The prime-indexing method is really cool, but it was already done to death in googological circles. I wanted to do something new... or at least, something that was new to me.

I also wanted my post to be understandable by the average beginner, which is why I limited my function to w^2. These are still mind-bogglingly huge numbers as far as most people are concerned (something that seasoned googologists often tend to forget, given how easy it is to generate such numbers in the FGH).