The reason k must be odd for base 2 is that if k is even, you can always divide it by 2 until you get an odd k, and increase n accordingly. Ex. 10*2^2+1=5*2^3+1. This simply eliminates testing the same number multiple times, and provides for a common format for these numbers. As far as n=0, I really think this should not be included for the following two reasons. First, it is not included in the original Sierpinski numbers, which we are trying to represent in base 5. Second, including n=0 eliminates all information on the base. For example, 4*2^0+1=4*5^0+1=4*45569^0+1. This defeats the purpose by reducing the expression k*b^n+1 to the much more general form k+1, or basically k.
