The number of ones in the binary representation of $3^n$ seems to continually increase as $n$ gets larger. Is there a known lower bound or is this still an unsolved problem? Looking at $n < 270.000$ the number of ones seems to always be more than $\frac{3}{4}n$ and grows at a pretty consistent rate. This data and the used rust program can both be found on github.
2026-03-30 11:58:44.1774871924
Is there a lower bound on the number of ones in the binary representations of powers of three?
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This sequence is A011754 in the OEIS. There the following conjecture, which is based on the heuristic argument that TonyK explained nicely in the comments,, is mentioned:
$$\lim_{n\to\infty} \frac{a_n}n = \frac{\log_2(3)}2>\frac34.$$
Since no paper on this subject is given at the OEIS, I suppose that nothing is known on this conjecture yet.