3. There are $n$ people, and it is known that any 2 of them have communicated at most once. Any $n-2$ of them have communicated the same number of times, which is equal to $3^{k}$ ($k$ is a positive integer). Find all possible values of $n$.


Obviously, $n \geqslant 5$. On one hand, since there are $C_{n}^{n-2}$ groups of $n-2$ people, and each group has a call frequency of $3^{k}$, the repeated count of call frequencies is $C_{n}^{n-2} \cdot 3^{k}$; on the other hand, each call between any two people $A, B$ is counted in $C_{n-2}^{n-4}$ groups of $n-2$ people, meaning the repeated count is $C_{n-2}^{n-4}$. Let the total number of calls be $m$, then
$$
\begin{array}{c}
C_{n-2}^{n-4} \cdot m=C_{n}^{n-2} \cdot 3^{k} \Rightarrow m=\frac{3^{k} n(n-1)}{(n-2)(n-3)} . \\
\text { When } 3 \nmid n \Rightarrow\left(n-3,3^{k}\right)=1,(n-3, n)=1 \Rightarrow n-3|n-1 \Rightarrow n-3| 2
\end{array}
$$
$$
\begin{array}{l}
\Rightarrow n=4 \text { or } 5 \Rightarrow n=5 \text {. } \\
\text { When } 3\left|n \Rightarrow\left(n-2,3^{k}\right)=1,(n-2, n-1)=1 \Rightarrow n-2\right| n \Rightarrow n-2 \mid 2
\end{array}
$$
$\Rightarrow n=3$ or 4, which is a contradiction.
When $n=5$, as long as any two people make a call once, it meets the requirement, so $n=5$.