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

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).