II. (40 points) Given an arithmetic sequence of integers $a_{1}, a_{2}, \cdots$, $a_{n}(n \geqslant 2)$ that satisfies $k \mid a_{k}(k=1,2, \cdots, n-1)$, and $n \nmid a_{n}$. Determine all possible values of the positive integer $n$.

Determine all possible values of the positive integer $n$.
When $n=p^{\alpha}$ ($p$ is a prime, $\alpha \in \mathbf{Z}_{+}$), let the least common multiple of $1, 2, \cdots, n-1$ be $C$, then take $a_{i}=C$ $(i=1,2, \cdots, n)$. Thus, for all $k=1,2, \cdots, n-1$, it always satisfies $k \mid a_{k}$, and the number of times the prime factor $p$ appears in $C$ is less than $\alpha$. Therefore, $n \nmid a_{n}$. Hence, such $n$ all meet the requirements.

When $n=p q(p, q>1, (p, q)=1)$, let the common difference of the arithmetic sequence $a_{1}, a_{2}, \cdots, a_{n}$ be $d$, and $p\left|a_{p}, p\right| a_{q}$.
$$
\begin{array}{l}
\text { and } a_{p q}=a_{p}+(p q-p) d, \\
a_{p q}=a_{q}+(p q-q) d,
\end{array}
$$

we know $p\left|a_{p q}, q\right| a_{p q}$.
Since $(p, q)=1 \Rightarrow p q\left|a_{p q} \Rightarrow n\right| a_{n}$, which contradicts the given requirement.
Therefore, all possible values of $n$ are $n=p^{\alpha}\left(p\right.$ is a prime, $\left.\alpha \in \mathbf{Z}_{+}\right)$.