Help with a prime number spiral which turns 90 degrees at each prime

I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the community. I will try to explain the puzzle as best I can then detail the questions I am interested in after. Imagine squared paper. In one square write the number $1.$ Continue to write numbers from left to right (as normal) until you reach a prime. The next number after a prime should be written in the square located $90$ degrees clockwise to the last. You then continue writing numbers in that direction. This procedure should be continued indefinitely. Here is a sample of the grid: $$\begin{array}{} 7&8&9&10&11&40&41 \\6&1&2&&12&&42\\5&4&3&14&13&44&43\\&&34&&26\\&&33&&27\\&&32&&28\\&&31&30&29\end{array}$$ Note that the square containing 3 also contains 15 (I couldn't put it in without confusing the diagram. In fact some squares contain multiple entries. I would have liked to see an expanded version of the diagram. I originally thought of shading squares that contain at least one number. Questions Does the square surrounded by $2,3,9,10,11,12,13,14$ ever get shaded? If so, will the whole grid ever be shaded? Is there a maximum number of times a square can be visited? I have got to 4 times but it is easy to make mistakes by hand. Are there any repeated patterns in the gaps? I have other ideas but this is enough for now as I have genuinely no idea how easy or difficult this problem is. Please forgive me for not taking it any further as it is so easy to make mistakes. I hope this is interesting for the community and look forwards to any results. Thanks. Any questions I'll do my best to clarify. Side note: I observed that initially at least the pattern likes to cling to itself but I suspect it doesn't later on.