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Q: Number of $1$'s in the ternary expansion of $a_p=\frac{3^{p-1}-1}{p}$

tanjiaI am considering a problem concerning Cantor set. Let $p>3$ be a prime. Is there a way to caculate the number of $1$ in the ternary expansion of $a_p=\frac{3^{p-1}-1}{p}$ or just to decide whether $1$ appears in the ternary expansion? Any result related to this quesion is also welcome.

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Q: Determining whether a natural number has no "1"s in its ternary expansion, without direct calculation

tanjiaI am considering a problem concerning Cantor set. Here is my quesion: Let $N\in\mathbb{N}$. Is there any way other than direct calculation to determine whether $N$ has no "1" as digits in its ternary expansion?

The tag was created and removed in 2012: mathoverflow.net/posts/83904/revisions
 

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