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195 lines
5.2 KiB
Python
Executable file
195 lines
5.2 KiB
Python
Executable file
"""#
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### 谜题描述
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Let's call the following process a transformation of a sequence of length n.
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If the sequence is empty, the process ends. Otherwise, append the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) (GCD) of all the elements of the sequence to the result and remove one arbitrary element from the sequence. Thus, when the process ends, we have a sequence of n integers: the greatest common divisors of all the elements in the sequence before each deletion.
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You are given an integer sequence 1, 2, ..., n. Find the lexicographically maximum result of its transformation.
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A sequence a_1, a_2, …, a_n is lexicographically larger than a sequence b_1, b_2, …, b_n, if there is an index i such that a_j = b_j for all j < i, and a_i > b_i.
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Input
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The first and only line of input contains one integer n (1≤ n≤ 10^6).
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Output
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Output n integers — the lexicographically maximum result of the transformation.
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Examples
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Input
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3
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Output
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1 1 3
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Input
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2
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Output
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1 2
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Input
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1
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Output
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1
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Note
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In the first sample the answer may be achieved this way:
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* Append GCD(1, 2, 3) = 1, remove 2.
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* Append GCD(1, 3) = 1, remove 1.
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* Append GCD(3) = 3, remove 3.
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We get the sequence [1, 1, 3] as the result.
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Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
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```python
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import sys
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def solve(n):
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if n == 1:
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return [1]
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elif n == 0:
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return []
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elif n == 3:
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return [1, 1, 3]
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res = [1 for _ in range(0, (n + 1) / 2)]
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res2 = solve(n - (n + 1) / 2)
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return res + map(lambda a: a * 2, res2)
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def gcd(a, b):
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if a > b:
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return gcd(b, a)
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if a == 0:
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return b
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return gcd(b % a, a)
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def read_ints():
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return map(lambda a: int(a), sys.stdin.readline().split())
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n, = read_ints()
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res = solve(n)
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print(\" \".join(map(lambda a: str(a), res)))
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```
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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。
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"""
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from bootcamp import Basebootcamp
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import re
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from bootcamp import Basebootcamp
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class Csequencetransformationbootcamp(Basebootcamp):
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def __init__(self, n=None, min_n=1, max_n=10):
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"""
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初始化谜题参数，可指定固定的n，或随机生成n的范围。
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参数:
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n (int, optional): 固定的n值。若提供，则生成的实例均为该n。
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min_n (int): 随机生成n时的最小值。
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max_n (int): 随机生成n时的最大值。
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"""
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self.n = n
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self.min_n = min_n
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self.max_n = max_n
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def case_generator(self):
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"""
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生成谜题实例，返回包含n值的字典。
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"""
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import random
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if self.n is not None:
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return {'n': self.n}
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else:
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n = random.randint(self.min_n, self.max_n)
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return {'n': n}
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@staticmethod
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def prompt_func(question_case) -> str:
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"""
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将谜题实例转换为详细的文本问题，指导用户按指定格式回答。
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"""
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n = question_case['n']
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prompt = f"""你是一个编程竞赛选手，需要解决以下问题：
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给定一个初始序列1, 2, ..., {n}，通过以下过程生成结果序列：
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1. 计算当前序列的GCD并添加到结果末尾。
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2. 删除序列中的一个元素。
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重复上述步骤直到序列为空，要求找到字典序最大的结果序列。
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输入：n = {n}
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输出：{n}个由空格分隔的整数，表示字典序最大的结果。
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示例：
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当n=3时，正确输出为：1 1 3，过程如下：
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1. GCD(1,2,3)=1，删除2 → 剩余[1,3]
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2. GCD(1,3)=1，删除1 → 剩余[3]
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3. GCD(3)=3，删除3 → 结果[1,1,3]
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请将答案放在[answer]标签内，例如：[answer]1 1 3[/answer]。
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你的任务是解决n={n}的情况，并将最终答案按指定格式放置。"""
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return prompt
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@staticmethod
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def extract_output(output):
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"""
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从模型输出中提取最后一个[answer]标签内的内容，并标准化格式。
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"""
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matches = re.findall(r'\[answer\](.*?)\[/answer\]', output, re.DOTALL)
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if not matches:
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return None
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last_match = matches[-1].strip()
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normalized = ' '.join(last_match.split())
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return normalized
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@classmethod
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def _verify_correction(cls, solution, identity):
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"""
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验证用户答案是否与正确结果一致。
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"""
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n = identity['n']
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try:
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correct = cls.solve(n)
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correct_str = ' '.join(map(str, correct))
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user_str = solution.strip()
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user_str = ' '.join(user_str.split())
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return user_str == correct_str
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except Exception:
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return False
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@staticmethod
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def solve(n):
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"""
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递归生成字典序最大的结果序列。
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"""
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if n == 1:
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return [1]
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elif n == 0:
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return []
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elif n == 3:
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return [1, 1, 3]
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res_length = (n + 1) // 2
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res = [1] * res_length
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remaining = n - res_length
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res2 = Csequencetransformationbootcamp.solve(remaining)
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res2_doubled = [x * 2 for x in res2]
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return res + res2_doubled
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