Powers of 2 whose Digits are Powers of 2

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Open Question

The only known powers of $2$ whose digits are also all powers of $2$ are:

$1, 2, 4, 8, 128$

This sequence is A130693 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Are there any more?


Demonstrated up to at least $2^{10 \, 000 \, 000}$.

This can be achieved by looking at the lowest $20$ digits only, by calculating those of successive powers of $2$ after applying the modulo operation.