Ping!

This might seem daunting at first, but it's not bad. Regardless of whether we see a ping at 0 or not, we cannot unambiguously find a satellite until we find SOME index that has a 1. The lowest non 0 index that has a 1 must have a satellite, and the index MUST be the interval.

Suppose the lowest index (i) that has a 1 ISN'T the interval of the satellite. That would mean there is some factor (f) of i which has a satellite. Since we haven't seen f before, it means there is some factor (f') of f which must have a satellite. This goes all the way back until we have some f' which is prime. For us to have not seen that, it would mean there would have had to be a ping at 1. This violates our assertion that i is the lowest index that we saw a 1 at. Reductio ad absurdum.

So anyway, now that we found a satellite (i) unambiguously, we want to remove it's effect from the output. So go through every multiple of i (k*i) and flip the bit. Now we can just repeat the whole algorithm again.