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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). enter image description here

enter image description here

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    Sorry! I mean two consecutive powers of 2 in their binary expansions – 2017-02-13
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    Yes, that edit make the question meaningful. – 2017-02-13
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    Most likely, your conjecture is true, but if so, proving it is not likely to be any easier than for a proof of the Erdos conjecture (assuming that's true as well). – 2017-02-13

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