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201 lines
5.2 KiB
Python
Executable file
201 lines
5.2 KiB
Python
Executable file
"""#
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### 谜题描述
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Let f(i) denote the minimum positive integer x such that x is not a divisor of i.
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Compute ∑_{i=1}^n f(i) modulo 10^9+7. In other words, compute f(1)+f(2)+...+f(n) modulo 10^9+7.
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Input
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The first line contains a single integer t (1≤ t≤ 10^4), the number of test cases. Then t cases follow.
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The only line of each test case contains a single integer n (1≤ n≤ 10^{16}).
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Output
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For each test case, output a single integer ans, where ans=∑_{i=1}^n f(i) modulo 10^9+7.
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Example
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Input
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6
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1
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2
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3
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4
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10
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10000000000000000
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Output
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2
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5
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7
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10
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26
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366580019
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Note
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In the fourth test case n=4, so ans=f(1)+f(2)+f(3)+f(4).
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* 1 is a divisor of 1 but 2 isn't, so 2 is the minimum positive integer that isn't a divisor of 1. Thus, f(1)=2.
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* 1 and 2 are divisors of 2 but 3 isn't, so 3 is the minimum positive integer that isn't a divisor of 2. Thus, f(2)=3.
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* 1 is a divisor of 3 but 2 isn't, so 2 is the minimum positive integer that isn't a divisor of 3. Thus, f(3)=2.
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* 1 and 2 are divisors of 4 but 3 isn't, so 3 is the minimum positive integer that isn't a divisor of 4. Thus, f(4)=3.
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Therefore, ans=f(1)+f(2)+f(3)+f(4)=2+3+2+3=10.
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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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import os
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from io import BytesIO
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sys.stdin = BytesIO(os.read(0, os.fstat(0).st_size))
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f = sys.stdin
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line = lambda: f.readline().strip('\r\n').split()
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def solve():
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idx = 0
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off = 1
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s = N
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mul = 1
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while mul <= N:
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mul = 1
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for i in range(idx,-1,-1):
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f = FACT[i]
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if mul % f != 0:
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mul *= f
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start = mul
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step = mul
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d = (N-start)//step + 1
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off = 1 if idx == 0 else FACT[idx+1] - FACT[idx]
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s += d * off
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s %= M
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idx += 1
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return str(s)
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FACT = [1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43]
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M = 10**9 + 7
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T = int(line()[0])
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for test in range(1,T+1):
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N = int(line()[0])
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sys.stdout.write(solve())
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sys.stdout.write(\"\n\")
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f.close()
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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 random
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import re
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from bootcamp import Basebootcamp
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class Cstrangefunctionbootcamp(Basebootcamp):
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def __init__(self, min_n=1, max_n=10**16, include_boundary=True):
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if min_n < 1:
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raise ValueError("min_n must be at least 1")
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if max_n < min_n:
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raise ValueError("max_n must be greater than or equal to min_n")
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self.min_n = min_n
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self.max_n = max_n
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self.include_boundary = include_boundary
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def case_generator(self):
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# 生成具有全面覆盖的测试用例
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if self.include_boundary and random.random() < 0.15:
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boundary_n = [1, 2, 3, 4, 10, 10000000000000000]
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n = random.choice(boundary_n)
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else:
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r = random.random()
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if r < 0.3: # 30%小值
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n = random.randint(1, 100)
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elif r < 0.6: # 30%中等值
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n = random.randint(10**3, 10**6)
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elif r < 0.85: # 25%大值
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n = random.randint(10**9, 10**12)
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else: # 15%极值
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n = random.randint(10**15, self.max_n)
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n = max(self.min_n, 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):
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n = question_case['n']
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return f"""请计算∑f(i) mod 1e9+7（1≤i≤{n}），其中f(i)是i的最小非因数。
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规则说明：
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1. f(i)定义为i的最小非除数正整数
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2. 例如f(4)=3，因为1、2是4的因数但3不是
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3. 总和需要对1e9+7取模
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输出要求：
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将最终答案放在[answer]标签内，如[answer]123[/answer]"""
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@staticmethod
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def extract_output(output):
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# 增强型数字提取逻辑
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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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number_str = re.sub(r'[^0-9]', '', matches[-1].strip())
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try:
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return int(number_str) % (10**9+7) if number_str else None
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except:
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return None
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@classmethod
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def _verify_correction(cls, solution, identity):
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try:
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return solution == cls.compute_answer(identity['n'])
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except:
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return False
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@staticmethod
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def compute_answer(n):
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# 优化后的参考实现
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FACT = [1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43]
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MOD = 10**9+7
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res = n % MOD
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idx = 0
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while idx < len(FACT):
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# 计算当前阶乘基
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product = 1
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for f in reversed(FACT[:idx+1]):
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if product % f != 0:
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product *= f
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if product > n:
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break
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if product > n:
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break
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# 计算贡献值
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offset = FACT[idx+1] - FACT[idx] if idx > 0 else 1
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count = (n - product) // product + 1
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res = (res + count * offset) % MOD
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idx += 1
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return res
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