1. The number of proper subsets of the set $\left\{n \left\lvert\,-\frac{1}{2}<\log _{\frac{1}{n}} 2<-\frac{1}{3}\right., n \in N\right\}$ is ().
(A) 7
(B) 8
(C) 31
() 32
$$
\begin{array}{l}
-\sqrt{1}(\mathrm{~A}) \\
-\frac{1}{2}<\log _{\frac{1}{n}} 2<-\frac{1}{3} \\
\Leftrightarrow \log _{\frac{1}{n}}\left(\frac{1}{n}\right)^{-\frac{1}{2}}<\log _{\frac{1}{n}} 2<\log _{\frac{1}{n}} \cdot\left(\frac{1}{n}\right)^{-\frac{1}{3}}, \\
\log _{\frac{1}{n}} \sqrt{n}<\log _{\frac{1}{2}} 2<\log _{5}-\sqrt[3]{n}, \\
\sqrt[2]{n}<2<\sqrt{n} .
\end{array}
$$

Solving, we get $4<n<8$.
Also, $n \in \mathbb{N}, n=5,6,7$. Thus, the original set is $\{5,6,7\}$, and the number of its proper subsets is $2^{3}-1=7$ (sets).
$$