Thanks, but the first reply really did make it all click together for me. I know just enough about number theory to know that this question was in the area of number theory, so I apologize that I don't know the correct terminology/notation, but my logic skills are quite solid. Had you not...
I stumbled on something while working on a program earlier; I'm not so arrogant to think it's never been noticed before, and I'm not sure it's even remotely useful, but I'm curious now and, assuming this has been explored pretty thoroughly, wanted to read more about it. Googling equations...