1.2.3 ** Let $P=\left\{n \mid n \geqslant 3, n \in \mathbf{N}_{+}\right\}$. The function $f: P \rightarrow \mathbf{N}_{+}$ is defined as follows: for $n \in P$, $f(n)$ is the smallest positive integer that is not a divisor of $n$. Find the range of the function $f$.
Parse the range $M=\left\{q \mid q \in \mathbf{N}_{+}, q\right.$ is a positive integer power of some prime $\}$ of the function $f$.
On one hand, let $q \in M$, i.e., there exists $n \in P$ such that $1,2, \cdots, q-1$ are all divisors of $n$, but $q \nmid n$. If $q$ is not a positive integer power of some prime, then $q$ can be decomposed into the product of two coprime positive integers $q_{1}$ and $q_{2}$, where $2 \leqslant q_{1}p^{\alpha}, 1 \leqslant i \leqslant k$. Let $n=p_{1}^{\alpha} \cdots$ $p_{k^{2}}^{\alpha_{k}} \cdot p^{\alpha-1}$, then by the definition of $f(n)$, we know that $f(n)=p^{\alpha}$.
In summary, the range of $f$ is $M$.