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This is an old unsolved problem. Erdos conjectured that for all $n\ge 9$ the ternary expansion of $2^n$ contains the ternary digit $2$ (this is equivalent to for every $n\ge 10$ the ternary expansion of $2^n$ contains a $1$). For recent work related to this (and references) see these papers of Abram and Lagarias, and Lagarias.