14.4.1 * Find all integers $k \geqslant 0$, such that the sequence $k+1, k+2, \cdots, k+10$ contains the maximum number of prime numbers.
When $k=1$, the sequence $k+1, k+2, k+3, \cdots, k+10$ (1) contains 5 prime numbers: 2, 3, 5, 7, 11. When $k=0, k=2$, the sequence (1) only contains 4 prime numbers. If $k \geqslant 3$, then the sequence (1) does not contain the prime number 3. It is known that in any three consecutive odd numbers, one is divisible by 3. Therefore, the sequence (1) contains at least one odd composite number. In addition, the sequence (1) contains 5 even composite numbers (since $k \geqslant 3$). Thus, when $k \geqslant 3$, the sequence (1) contains at least 6 composite numbers. Hence, the number of prime numbers does not exceed 4. The required $k$ is 1.