Sum on digits of powers of two is not too large

Question

Is the following proved:

Are there infinitely many positive integers $m$ and an integer $n$ such that sum of digits of $2^m$ is at most $n$?

Answer 1

This is too long to be a comment.

Let $\phi(m)=\text{sum of digits of }m$. You ask if $\liminf_{m\to\infty}\phi(m)<\infty$. $2^m$ has $m\,\log_{10}2$ digits. The mean of the $9$ digits is $4.5$. We expect $$ \phi(m)\sim4.5\,\log_{10}2\,n=1.35463498\,m. $$

I have computed $\phi(m)$ for $1\le m\le 100000$. The mean value of $\phi(m)/m$ is $1.3548$ with variance $0.000244$, in agreement with the above. Here is the histogram: