11. (2004-2005 Hungarian Mathematical Olympiad) Let $n$ be a positive integer. If there exist integers $a_{1}$, $a_{2}, \cdots, a_{n}$ (not necessarily distinct) such that $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=n$, then $n$ is called "charming." Find the charming integers.
11. If $k=4 t+1, t \in \mathbf{N}$, it clearly meets the requirement. Take $4 t+1$ and $2 t$ ones, $2 t$ negative ones.
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If $k=4$, then $a_{1} a_{2} a_{3} a_{4}=4$. It can only be $a_{4}=4$ or $a_{4}=a_{3}=2$, which clearly has no solution.
If $k=4 t, t \geqslant 2$, consider two cases.
When $t$ is odd, take $2 t, -2, x$ ones, $y$ negative ones ($x, y$ to be determined), then $\left\{\begin{array}{l}x+y=4 t-2, \\ x-y+2 t-2=4 t\end{array}\right.$.
Solving, we get $x=3 t, y=t-2$.
Clearly, this set of numbers meets the requirements.
When $t$ is even, similarly take $2 t, 2, x$ ones, $y$ negative ones ($x, y$ to be determined), then $\left\{\begin{array}{l}x+y=4 t-2, \\ x-y=2 t-2 .\end{array}\right.$
Solving, we get $x=3 t-2, y=t$.
This set of numbers must meet the requirements.
In summary, numbers of the form $4 t (t \geqslant 2)$ are charming.
Next, we prove: numbers of the form $4 t+2, 4 t+3$ are not charming.
If numbers of the form $4 t+2$ are charming, let
$4 t+2=a_{1}+a_{2}+\cdots+a_{4 t+2}=a_{1} a_{2} \cdots a_{4 t+2}$.
It is easy to see that among $a_{i}$, there is exactly one even number, and the remaining $4 t+1$ numbers are odd, so $a_{1}+a_{2}+\cdots+a_{4 t+2}$ must be odd. Contradiction.
Therefore, numbers of the form $4 t+2$ are not charming.
If numbers of the form $4 t+3$ are charming, let
$4 t+3=a_{1} a_{2} \cdots a_{4 t+3}=a_{1}+a_{2}+\cdots+a_{4 t+3}$, where there are $x$ numbers congruent to 1 modulo 4, and $4 t+3-x$ numbers congruent to 3 modulo 4. Hence $x+3(4 t+3-x) \equiv 3(\bmod 4)$.
So, $2 x \equiv 2(\bmod 4)$.
Thus, $x$ is odd, and $4 t+3-x$ is even.
Then $3 \equiv 4 t+3=a_{1} a_{2} \cdots a_{4 t+3} \equiv 1^{x} \times 3^{4 t+3-x} \equiv 1(\bmod 4)$.
Contradiction.
Therefore, numbers of the form $4 t+3$ are not charming.
In summary, all charming numbers are
$$
4 t+1, t \in \mathbf{N} ; 4 t, t \in \mathbf{N}, t \geqslant 2 .
$$