Determine all positive integers $n$ for which $B_n=\{0\}$.

Determine all positive integers $n$ for which $B_n=\{0\}$.

Let $A_1,A_2,...,A_n,...$ and $B_1,B_2,...,B_n,...$ be sequences of sets
defined by $a_1=\emptyset$, $B_1=\{0\}$, $A_{n+1}=\{x+1|x\in
B_n\},B_{n+1}=(A_n\cup B_n)\(A_n\cap B_n)$. Determine all positive
integers $n$ for which $B_n=\{0\}$.
I think this problem is from a Chinese Math Olympiad. Also by
experimentation the answer seems to be all powers of 2.