Erdos conjectured that, except $1, 4$ and $256$, no power of $2$ is a sum of distinct powers of $3$.
A vice-versa conjecture may be: except $1$, $9$ and $81$ all powers of $3$ contains two consecutive powers of $2$ in theirs binary expansions?
We can generate the powers $(3^n)$ as a cellular automata where the n-th line represent the binary expansion of $3^n$ and we can observe that the conjecture still verified as showed in the next images (with respectively 128 lines and 1024 lines).
