Okay, so the charts are filled out as fully as I’m filling them out tonight, and I promised an explanation, so this is a quick one. (It’s late.)
I took the interesting number above, 101010101, which has more than two prime factors, and looked at the results of my characterize.pl script, which looked at remainders when m=101010101 subtracts from its Ceiling Square, the square of (its Ceiling Root plus two), the square of (its Ceiling Square plus four), and the square of (its Ceiling Square plus six). I did the same with an actual nearby discrete semiprime, 101161639. I then took those remainders for each number, which were not perfect squares, and looked at the differences between them and their own Ceiling Squares. I used that information to fill out a table. I’ll be fleshing out spaces between the rows and columns of the tables I have, to see if properties emerge, since the rows and columns of numbers each satisfy quadratic polynomials.
I will explain more fully what I’m doing if it actually leads me somewhere useful. It might! Who knows? We’ll just have to see.