if we marked sequences of integers with 3 options. even, odd, other. then these lists are not finite in bases of 3^k.

for four options. even, odd, other, another. then these lists are not finite in bases of 4^k.

there is an intersection in the infinite lists where the base is equivalent to the power of an earlier base.

so infinite lists for 2^k would overlap a subset of the infinite lists for 2^2^k=4^k

all prime bases, p, p^k would admit infinite lists that cover all the infinite lists for some composite base, c, c^k.