Powers of 2 whose Digits are Powers of 2

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?

Progress

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.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $128$
• 1994: Saunders: No more powers of 2 with this property up to $2^{70000}$ (J. Recr. Math. Vol. 26: p. 151)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $128$