[madcaptenor]

Looks like that's all of them. The typical number of even digits of n grows like a constant times n, so you need some very large deviations from t

I'd conjecture the number of powers of 2 with exactly m even digits is finite for all m.

[esnard]

Just tested up to 2 * 1000000, and it indeed looks like that's all of them.

[recursive]

Asterisk means exponent here?

[esnard]

Yes. I didn't escape the asterisks before posting, and HN replaced ** with \* without me noticing.