Does there exist an integer $n$ such that the decimal representation of $2^n$ have an equal number of decimal digits $\{0,\dots,9\}$, each appearing 10% of the time?
The closest I could find was $n=1,287,579$ of which $2^n$ has 387,600 digits broken down as
0 38,808 10.012%
1 38,735 9.993%
2 38,786 10.007%
3 38,751 9.997%
4 38,814 10.014%
5 38,713 9.987%
6 38,731 9.992%
7 38,730 9.992%
8 38,709 9.986%
9 38,823 10.016%
Answer
No. If each digit appears $x$ times, then the sum of all the digits will be $45x$; this implies $3|2^n$ which cannot be the case.
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