Perfect power of $2$ in whose decimal representation, each digit is also a non-negative power of $2$

Question

Let $A:=\{n \in \mathbb N : 2^n=\sum_{j=0}^k 10^j2^{n_j} , $ for some integers $ 0 \le n_j \le 3 , j=0,...,k $ with $ k>1\}$ ; then is the set $A$ finite or infinite ? I have been unable to find any number other than $7$ in $A$ . Please help . Thanks in advance .