# When does $2^n$ start with n?

A trick my dad taught me for easily referencing powers of 2 is that $2^6=64$ and $2^{10}=1024$, because $64$ starts with $6$ and $1024$ starts with $10$, and so it's faster than manually doubling numbers if I wanted to find $2^7$ or something quickly in a math competition.

This made me curious to think - does this property that $2^n$ begins with $n$ in its base 10 representation hold for any other $n$'s besides 6 and 10? How about in other bases? What kind of mathematics would this type of problem fall under, and how would one go about solving it?

• $2^{1542}$ and $2^{77075}$ have this property May 30, 2017 at 4:57
• @user170039 Huh? May 30, 2017 at 4:58
• the binary representation of $2^n$ starts with $n$ whenever $n$ itself is a power of 2
– WW1
May 30, 2017 at 5:01
• @user170039 Sorry, I still have no idea why you would ask this. May 30, 2017 at 5:10
• More examples found at oeis.org/A100129 May 30, 2017 at 5:41