I was hoping that there would be a way of bringing out all the exclusion lines at the same time... and I suddenly thought, why not start the Ulam spiral at zero... rather than 1.

By starting the Ulam spiral at zero and leaving the X and Y axis alone (no movement between iterations) things came together a little better.

Here is the result after 1 iteration (Standard Ulam spiral)....

... then after layering 10 iterations...

... and after 30 iterations ....

... and after 50 iterations ....

... and after 150 iterations ....

... and then after 350 iterations ....

... and then after 700 iterations ....

... and then after 2000 iterations ....

For someone who sat for hours drawing those lines by hand, thinking they were quite mad sometimes (ok, I am a bit mad for doing that anyway), this was a heartening moment.... the lines were drawing themselves! It was quite amazing to watch this sequence evolve in real time, Iteration by iteration.

Below is an animation of the first 1000 iterations... YouTube has compressed and changed the aspect ratio a bit, but you can still see the effect.