JCSBimp's Hexagonal Orthogonal

I am working several ideas, and haven’t dropped any of the recent ones I’ve started examining. The weeds/woods are deep and thick. It’s all worthwhile. Plus, I woke up before my Sunday alarm today from dreams that played with the Fermat difference-of-squares approach and my current algorithm that uses it. MIRIAM is still with me. That’s good.

Now I am thinking along the lines of lining up odd integers according to their remainders when subtracted from their Ceiling Squares, since there is of course the natural lining-up that occurs when those remainders are themselves perfect squares. There are n odd integers between n² and (n+1)², with the latter being their Ceiling Square, and one can tick off and eliminate the perfect square remainder cases with ease. Lining the others up, is there anything we can observe? One can start with the smallest possible m having remainder r when the Ceiling Square is subtracted. What are their properties, and are there any useful interactions they could have with one another?

I love that this integer factorization study has not yet failed to give me something else interesting to do, and to try.