I got interested in the Ulam Spiral a while back. I’m not sure what brought me to the Wikipedia page but it instantly grabbed my attention. First off I am not a mathematician so some of this may have already been thought of and looked at. I have searched and not found what I have come up with. I also don’t write like a mathematician so if my formulas break convention and they probably do that is just going to be the way it is.
If you have come here and don’t know what the Ulam spiral is I highly recommend the Wikipedia page on it.
The first thing that I did was draw one for myself. I always find that doing something step by step rather than having someone else or a program do it helps with understand it. These first ones were all in my note book and were quite small. The image above is from later in my thought process.