A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
1. **Understanding the Problem:**
   We need to find all positive integers \( n \) that can be uniquely written as both a sum and a product of positive integers \( a_1, a_2, \ldots, a_k \) where \( k \geq 2 \). This means:
   \[
   n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdot \cdots \cdot a_k
   \]
   and this representation must be unique.

2. **Analyzing the Example:**
   Consider \( n = 10 \):
   \[
   10 = 5 + 2 + 1 + 1 + 1 = 5 \cdot 2 \cdot 1 \cdot 1 \cdot 1
   \]
   This is a unique representation of 10 as both a sum and a product.

3. **Considering Prime Numbers:**
   Prime numbers cannot be good because they cannot be expressed as a product of two or more positive integers other than 1 and themselves. Hence, primes are not good natural numbers.

4. **Considering Composite Numbers:**
   Let \( n = ij \) where \( i, j \neq 1 \) are positive integers. Then:
   \[
   n = i + j + 1 + 1 + \cdots + 1 \quad \text{(there are \( n - i - j \) ones)}
   \]
   \[
   n = i \cdot j \cdot 1 \cdot 1 \cdot \cdots \cdot 1
   \]
   This representation is denoted as \( A(i, j) \).

5. **Uniqueness Condition:**
   For \( n \) to have a unique representation, it must be uniquely expressible as a product of two positive integers (both not equal to 1). This implies:
   \[
   n = p^2 \quad \text{or} \quad n = pq
   \]
   where \( p \) and \( q \) are primes.

6. **Verification:**
   - If \( n = p^2 \), then:
     \[
     n = p + p = 2p \quad \text{and} \quad n = p \cdot p = p^2
     \]
     This is a unique representation.
   - If \( n = pq \), then:
     \[
     n = p + q \quad \text{and} \quad n = p \cdot q
     \]
     This is also a unique representation.

7. **Conclusion:**
   Therefore, the good natural numbers are those that can be expressed as \( p^2 \) or \( pq \) where \( p \) and \( q \) are primes.

The final answer is \( \boxed{ n = p^2 } \) or \( n = pq \) where \( p \) and \( q \) are primes.