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internbootcamp/bootcamp/cehabandaspecialcoloringproblem/cehabandaspecialcoloringproblem.py

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+"""# 
+
+### 谜题描述
+You're given an integer n. For every integer i from 2 to n, assign a positive integer a_i such that the following conditions hold:
+
+  * For any pair of integers (i,j), if i and j are coprime, a_i ≠ a_j. 
+  * The maximal value of all a_i should be minimized (that is, as small as possible). 
+
+
+
+A pair of integers is called [coprime](https://en.wikipedia.org/wiki/Coprime_integers) if their [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) is 1.
+
+Input
+
+The only line contains the integer n (2 ≤ n ≤ 10^5).
+
+Output
+
+Print n-1 integers, a_2, a_3, …, a_n (1 ≤ a_i ≤ n). 
+
+If there are multiple solutions, print any of them.
+
+Examples
+
+Input
+
+
+4
+
+
+Output
+
+
+1 2 1 
+
+Input
+
+
+3
+
+
+Output
+
+
+2 1
+
+Note
+
+In the first example, notice that 3 and 4 are coprime, so a_3 ≠ a_4. Also, notice that a=[1,2,3] satisfies the first condition, but it's not a correct answer because its maximal value is 3.
+
+Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
+```python
+resp = [0]*100001
+
+n = int(raw_input())
+count = 1
+for i in range(2, n+1):
+	if resp[i] == 0:
+		for j in range(i, n+1, i):
+			resp[j] = count
+		count += 1
+
+for i in range(2, n+1):
+	print resp[i],
+```
+
+
+请完成上述谜题的训练场环境类实现，包括所有必要的方法。
+"""
+
+from bootcamp import Basebootcamp
+import math
+from collections import defaultdict
+from itertools import combinations
+from math import gcd
+import re
+
+class Cehabandaspecialcoloringproblembootcamp(Basebootcamp):
+    def __init__(self, min_n=2, max_n=100):
+        self.min_n = min_n
+        self.max_n = max_n
+    
+    def case_generator(self):
+        import random
+        n = random.randint(self.min_n, self.max_n)
+        return {'n': n}
+    
+    @staticmethod
+    def prompt_func(question_case):
+        n = question_case['n']
+        prompt = f"""You are given an integer n = {n}. Your task is to assign a positive integer a_i to each integer i from 2 to n such that the following conditions are met:
+
+1. For every pair of coprime integers (i, j), a_i ≠ a_j.
+2. The maximum value among all a_i is as small as possible.
+
+Two integers are coprime if their greatest common divisor (GCD) is 1. Your solution should output the values a_2, a_3, ..., a_n separated by spaces.
+
+Examples:
+
+Input:
+4
+Output:
+1 2 1
+
+Input:
+3
+Output:
+2 1
+
+Your answer should include the sequence of numbers for a_2 to a_n enclosed within [answer] and [/answer] tags. For example, if your solution is "1 2 1" for n=4, write it as [answer]1 2 1[/answer]."""
+        return prompt
+    
+    @staticmethod
+    def extract_output(output):
+        matches = re.findall(r'\[answer\](.*?)\[/answer\]', output, re.DOTALL)
+        if not matches:
+            return None
+        last_match = matches[-1].strip()
+        try:
+            solution = list(map(int, last_match.split()))
+            return solution
+        except:
+            return None
+    
+    @classmethod
+    def _verify_correction(cls, solution, identity):
+        n = identity['n']
+        if len(solution) != n - 1:
+            return False
+        
+        def count_primes(k):
+            if k < 2:
+                return 0
+            sieve = [True] * (k + 1)
+            sieve[0] = sieve[1] = False
+            for i in range(2, int(math.sqrt(k)) + 1):
+                if sieve[i]:
+                    sieve[i*i : k+1 : i] = [False] * len(sieve[i*i : k+1 : i])
+            return sum(sieve)
+        primes_count = count_primes(n)
+        
+        max_color = max(solution) if solution else 0
+        if max_color != primes_count:
+            return False
+        
+        color_groups = defaultdict(list)
+        for idx, color in enumerate(solution):
+            i = idx + 2  # i is from 2 to n
+            color_groups[color].append(i)
+        
+        for group in color_groups.values():
+            for a, b in combinations(group, 2):
+                if gcd(a, b) == 1:
+                    return False
+        return True