195 lines
6.0 KiB
Python
195 lines
6.0 KiB
Python
"""
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By replacing the 1st digit of the 2-digit number *3,
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it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
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By replacing the 3rd and 4th digits of 56**3 with the same digit,
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this 5-digit number is the first example having seven primes among the ten generated numbers,
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yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993.
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Consequently 56003, being the first member of this family, is the smallest prime with this property.
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Find the smallest prime which,
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by replacing part of the number (not necessarily adjacent digits) with the same digit,
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is part of an eight prime value family.
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-----
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*.000*.、*.111*.、*.222*.、*.333*.、*.444*.、*.555*.、*.666*.、*.777*.、*.888*.、*.999*.
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数字的形式就是这样，最多十个数，那么，就是要在质数序列上搜索这样的数列:
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1. 用欧拉筛生成质数表（上限约100万足够）
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2. 对每个质数p：
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a. 检查是否有≥3个相同数字，且该数字∈{0,1,2}
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b. 生成所有C(n,3)种3位替换组合
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c. 对每种组合，用0-9替换，统计质数个数
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d. 若得到8个质数，返回p（首个即为答案）
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"""
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import itertools
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import time
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from collections import Counter
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from math import isqrt
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from bitarray import bitarray
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def timer(func):
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def wrapper(*args, **kwargs):
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start_time = time.time()
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result = func(*args, **kwargs)
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end_time = time.time()
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elapsed_time = end_time - start_time
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print(f"{func.__name__} time: {elapsed_time:.6f} seconds")
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return result
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return wrapper
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def replace_digit(
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number: str | int,
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target_digit: str | int,
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replace_count: int = 3,
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replacement_digit: str | int = "0123456789",
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return_positions: bool = False,
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) -> list[str] | list[tuple]:
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"""
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使用单个替换数字，替换指定数量的目标数字
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参数:
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number: 输入的数字（字符串或整数）
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target_digit: 要被替换的目标数字（如 '1'）
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replace_count: 要替换的数量（必须 <= 目标数字出现的次数）
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replacement_digit: 用于替换的数字（单个字符）
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return_positions: 是否同时返回被替换的位置信息
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返回:
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如果return_positions为False，返回替换后的数字列表
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如果return_positions为True，返回[(替换后的数字, 被替换的位置), ...]
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"""
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# 转换为字符串处理
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num_str = str(number)
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target = str(target_digit)
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replacement = str(replacement_digit)
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# 找出所有目标数字的位置
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target_positions = [i for i, digit in enumerate(num_str) if digit == target]
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# 检查是否有足够的目标数字
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if len(target_positions) < replace_count:
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return [
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f"错误：目标数字 '{target}' 只出现了 {len(target_positions)} 次，少于要替换的数量 {replace_count}"
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]
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results = []
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# 遍历所有可能的要替换的位置组合
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for positions in itertools.combinations(target_positions, replace_count):
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# 创建新数字的列表
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num_list = list(num_str)
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# 替换指定位置
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for pos in positions:
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num_list[pos] = replacement
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new_number = "".join(num_list)
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if return_positions:
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results.append((new_number, positions))
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else:
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results.append(new_number)
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return results
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def replace_muldigits(
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number: str | int,
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target_digit: str | int,
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replace_count: int = 3,
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replacement_digits: str | list[str] = "0123456789",
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return_positions: bool = False,
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) -> list:
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"""
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使用多个替换数字，每个数字单独进行替换
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参数:
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number: 输入的数字
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target_digit: 目标数字
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replace_count: 要替换的数量
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replacement_digits: 多个替换数字（字符串或列表）
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return_positions: 是否返回位置信息
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返回:
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字典，键为替换数字，值为对应的结果列表
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"""
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# 转换替换数字为列表
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if isinstance(replacement_digits, str):
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replacement_list = list(replacement_digits)
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else:
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replacement_list = replacement_digits
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results = {}
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for rep_digit in replacement_list:
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results[rep_digit] = replace_digit(
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number, target_digit, replace_count, rep_digit, return_positions
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)
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results = list(results.values())
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res = [[] for _ in range(len(results[0]))]
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for r in results:
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for i, val in enumerate(r):
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if val[0] != "0":
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res[i].append(int(val))
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return [sorted(list(set(val))) for val in res]
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def primes_list(limit: int = 10**8) -> list[int]:
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is_prime = bitarray(limit + 1)
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is_prime.setall(True)
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is_prime[:2] = False # 0和1不是素数
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# 只需筛到 sqrt(n)
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imax = isqrt(limit) + 1
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for i in range(2, imax):
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if is_prime[i]:
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# 步长i，从i*i开始（小于i*i的已被更小的素数筛过）
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is_prime[i * i : limit + 1 : i] = False
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return [i for i, isp in enumerate(is_prime) if isp]
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def have_n_primes(primes: list[int], target: list[int]) -> int:
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count = 0
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for n in target:
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if n in primes:
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count += 1
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return count
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@timer
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def solve() -> list[int]:
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primes = primes_list()
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for p in primes:
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if p < 1000:
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continue
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digits = [int(d) for d in str(p)]
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digit_counts = Counter(digits)
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if max(digit_counts.values()) < 3:
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continue
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digs = [k for k, v in digit_counts.items() if v >= 3 and k in [0, 1, 2]]
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for dig in digs:
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testList = replace_muldigits(p, dig)
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for xList in testList:
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if have_n_primes(primes, xList) == 8:
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return [x for x in xList if x in primes]
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return []
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if __name__ == "__main__":
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result = solve()
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print(result)
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