"""
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.

For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110; therefore d(220) = 284.
The proper divisors of 284 are 1, 2, 4, 71, and 142; so d(284) = 220.

Evaluate the sum of all the amicable numbers under 10000.
"""

import time


def timer(func):
    def wrapper(*args, **kwargs):
        start_time = time.time()
        result = func(*args, **kwargs)
        end_time = time.time()
        print(f"{func.__name__} took {end_time - start_time:.8f} seconds")
        return result

    return wrapper


def sieve_primes(limit):
    """埃拉托斯特尼筛法生成质数列表"""
    is_prime = [True] * (limit + 1)
    is_prime[0:2] = [False, False]
    primes = []
    for i in range(2, limit + 1):
        if is_prime[i]:
            primes.append(i)
            for j in range(i * i, limit + 1, i):
                is_prime[j] = False
    return primes


def prime_factors_with_sieve(n, primes):
    """使用预计算的质数列表进行分解"""
    factors = {}
    temp = n
    for p in primes:
        if p * p > temp:
            break
        while temp % p == 0:
            factors[p] = factors.get(p, 0) + 1
            temp //= p
    if temp > 1:
        factors[temp] = factors.get(temp, 0) + 1
    return factors


def get_divisors_optimized(n):
    """优化版本：预计算质数加速分解"""
    if n < 2:
        return [1] if n == 1 else []

    # 计算需要的质数范围
    limit = int(n**0.5) + 1
    primes = sieve_primes(limit)

    # 质因数分解
    factors = prime_factors_with_sieve(n, primes)

    # 生成除数
    divisors = [1]
    for p, exp in factors.items():
        powers = [p**e for e in range(exp + 1)]
        divisors = [a * b for a in divisors for b in powers]

    return sorted(divisors)


def sum_of_divisors(n):
    """计算一个数的所有真因子之和"""
    return sum(get_divisors_optimized(n)) - n


def find_amicable_numbers(limit):
    """查找小于给定上限的全部亲和数对"""
    amicable_pairs = set()
    for a in range(2, limit):
        b = sum_of_divisors(a)
        if a != b and sum_of_divisors(b) == a:
            amicable_pairs.add((min(a, b), max(a, b)))
    return amicable_pairs


@timer
def sum_of_amicable_numbers(limit):
    """计算小于给定上限的全部亲和数之和"""
    return sum(a + b for a, b in find_amicable_numbers(limit))


if __name__ == "__main__":
    print(sum_of_amicable_numbers(10000))