Sum of digits of 2^n equals n

[Joshua Zucker]

On 7/23/06, Tanya Khovanova <tanyakh at tanyakhovanova.com> wrote:
> I found two numbers n such that sum of digits of 2^n is equal to n:
> 5 and 70.
>
> I can't find the corresponding sequence.
>
> Are there any other numbers like that?

Interesting question!  Since log(2^n) = about 0.3n,
and each digit on average is about 4.5, (ignoring stuff like the
restrictions on the last digit, and that the first digit is more often
1 and less often 9, and so on),
you'd expect the digit sum on average to be about 1.35n,
and so a little too big.

So I wouldn't be surprised to find that 5,9,10,17,70 are the only ones
where the digit sum of 2^n is less than or equal to n, and 5 and 70
therefore the only ones with equality.

That's not a proof, of course, but having checked up to 10000 and not
found any more terms ...

--Joshua Zucker