The images in my last post were particular realizations of the famous Sieve of Eratosthenes. The basic idea of the sieve is to repeatedly do the following:
- Circle the next number bigger than
that is not yet crossed out, call it
.
- Cross out every
This is an efficient way to find all the primes up to a given limit. Note that it doesn’t require doing any division or factoring, just adding. Here’s the image of the 
Some questions for you to ponder:
- Why can we always cross out multiples of each prime
- When will the lines be vertical? When will they be diagonal?
- Is there only one way to cross out multiples of each
- Why are the primes on the top row circled, while the rest of the primes are in boxes? What’s the difference?
And just for fun here’s the sieve diagram for 
The Math Less Traveled 
Really fun! I now see that I could have taken my comment on the previous post a bit further: all lines go down-left if $n – 1$ is a multiple of $p$, which you already do, except for $2$.
I do wonder how you choose which line to draw: I thought it would be to draw the line through the closest (in distance) next multiple of $p$, but then the line for $p = 17$ in the drawing for $n = 29$ would have gone through $102$ instead of $51$.
My code only looks for slopes which are integers or reciprocals of integers (or vertical).
In more detail, think of the slope of a line as given in the form (dr, dc), that is, the offset in terms of rows and columns from one crossed out number to the next. Then I only consider (1, -n mod p) and (-n^{-1} mod p, 1) and pick whichever one has the smaller absolute value for the non-unit dimension. There are certainly other ways it could be done — I like your idea of picking the closest.
Hello. Can you give the code used to do the plot ?
A lazzy man. 😉
Sure, the code is available here: https://hub.darcs.net/byorgey/mathlesstraveled/browse/2019-10-sieve/Sieve.hs It ended up being a lot more complicated to code than I thought it would be.