1. 47 If a natural number $n$ can be uniquely represented as the sum $a_{1}+a_{2}+\cdots+a_{k}$ and the product $a_{1} a_{2} \cdots a_{k}$ of $k(k \geqslant 2)$ natural numbers, then this natural number is called a "good" number. For example, 10 is a "good" number, because $10=5+2+1+1+1=5 \times 2 \times 1 \times 1$ $\times 1$, and this representation is unique. Try to use the term prime to indicate which natural numbers are "good" numbers.
[Solution] If $n$ is a prime number, then $n$ is not a "good" number. Because $n=n \cdot 1<n+1$, such $n$ cannot be expressed as the sum and product of the same integers.

If $n=p \cdot q$, where $p$ and $q$ are both primes, then
$n=p q=p+q+1 \times(p q-p-q)$, and this representation is unique.
If $n=a b c$, where $a, b, c$ are all integers greater than 1, then
$$n=a b c=a b+c+1 \times(a b c-a b-c)$$
$=a+b+c+1 \times(a b c-a-b-c)$. Therefore, this representation is not unique. Hence, $n$ is a "good" number if and only if $n$ is the product of two prime numbers.