For base 2, you can use Pascal's triangle to find out how many numbers with x zeroes (or ones) exist up to 2^n: http://alquerubim.blogspot.com/2010/10/combinacoes-de-digito... (portuguese only, I'm afraid).

From base 3 onwards, I would expect to see some "subparticles" inside each number in the triangle. I would start from there.