Hi, whats an example of a subset A of natural numbers where the base 2
representation of A is a regular language(in the finite automata
sense) and the base 3 representation is a non-regular language and how
do you prove that the example works? Also, assume that correct base
representations of numbers don't have unnecessary leading zeros.

I tried doing it, but im stuck. What I have so far is that 2^n for all
n is a possible candidate and a possible string to use with the
pumping lemma is the base 3 representation of 2^(phi(3^p))-1 where phi
is the totient function because this yields a string with 0 p's on the
right. I dont know how to use the pumping lemma here though.