Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity
a) not greater than the complexity of $n$.
b) less than the complexity of $n$.

(Boris Frenkin)
1. **Define the complexity of \( n \)**:
   Let \( n = p_1 p_2 \cdots p_m \), where \( p_1 \leq p_2 \leq \cdots \leq p_m \) are prime factors of \( n \). The complexity of \( n \) is \( m \), the number of prime factors in its prime decomposition.

2. **Consider the range between \( n \) and \( 2n \)**:
   We need to find all \( n \) such that all integers between \( n \) and \( 2n \) have complexity:
   a) Not greater than the complexity of \( n \).
   b) Less than the complexity of \( n \).

3. **Analyze the prime factors**:
   Let \( n = t \cdot p_m \), where \( t = p_1 p_2 \cdots p_{m-1} \). Consider the numbers \( t \cdot (p_m + 1), t \cdot (p_m + 2), \ldots, t \cdot (p_m + p_m - 1) \). These numbers are between \( n \) and \( 2n \).

4. **Condition for \( p_m > 2 \)**:
   If \( p_m > 2 \), then \( p_m + 1 \) has complexity more than one. Therefore, \( t \cdot (p_m + 1) \) has complexity more than \( n \). This implies that \( n \) must be a power of two to satisfy the condition.

5. **Prove that any power of two satisfies condition (a)**:
   Suppose \( n = 2^k \). We need to show that all integers between \( 2^k \) and \( 2^{k+1} \) have complexity not greater than \( k \).
   - Assume there exists a number \( 2^k < k = p_1 p_2 \cdots p_m < 2^{k+1} \) with \( m \geq k \).
   - Since \( k \) cannot be a power of two, we have \( 2^k > k > 2^m \geq 2^k \), which is a contradiction.

6. **Prove that any power of two does not satisfy condition (b)**:
   - Consider the number \( 2^k < 2^{k-1} \cdot 3 < 2^{k+1} \).
   - The number \( 2^{k-1} \cdot 3 \) has complexity \( k \), which is not less than the complexity of \( 2^k \).

The final answer is \( \boxed{ n } \) must be a power of two for condition (a), and no \( n \) satisfies condition (b).