That is wierd. Base k includes numbers 0,...,k-1. For example 101 is a valid number in base 2 which is 2^2 + 2^1. Using that logic, base 1 would only contain 0

That is wierd. Base k includes numbers 0,...,k-1. For example 101 is a valid number in base 2 which is 2^2 + 2^1. Using that logic, base 1 would only contain 0, hence it should not be able to expresss more values that 0. Is your example actually a valid base 1?