[lanna]

How many powers of 2 have just a single even digit? 2, 4, 8, 16, 32, 512...

[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.

[sltkr]

This is equivalent to asking: how many powers of 2 are there such that all digits except the leading digit are 5 or greater?

The powers of 2 with a single even digit are just those double those numbers (i.e., the next higher power of 2).