Let $k \ne 0$ and $3\leq d_1 < d_2 < \cdots < d_r$ be integers of gcd equal to $1$ such that $$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$ Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i \in \{0, 1\}$ and $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when written in base $d_i$?

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