I 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.

Given a number $n$ and an Interval $I = [ \; \lfloor n^{1/4} \rfloor, \lfloor n^{(1/3) \rfloor \;} ]$, can we say anything about the distribution of $\{ n \mod b \;\;| \; b \in I \}$? In particular, if I wanted the residue to be close to any region in $[0, b-1]$, say close to the "top" of the re...

Let $n$ be the positive integer. Let $A$ and $B$ be sets of divisors of $n$ less and more than $\sqrt{n}$ respectively. Consider bipartite graph $(A, B)$, where two vertices are connected when one divides another. Denote $M(n)$ number of perfect matchings in this graph. Is $M(n) > 0$ for all $n$(...