INF2080 Challenge II

We have received multiple submissions for the last challenge, many of which were correct, some of which were extremely close! A sample solution will be posted some time this week. So on to the next challenge:

Let A be a set of natural numbers, and k a natural number greater then one. Define

B_k(A)= {w | w is the representation in base k, without leading zeros, of some number in A}.

For example: B₂({3,5})={11,101} and B₃({3,5})={10,12}. Give an example of a set A for which B₂(A) is regular and B₃(A) is not regular (with proof).