Project Euler 347

发表于 2022-06-22 更新于 2023-07-03 分类于 Project Euler 阅读次数： 91 本文字数： 873 阅读时长 ≈ 1 分钟

Project Euler 347

题目

Largest integer divisible by two primes

The largest integer $\leq 100$ that is only divisible by both the primes $2$ and $3$ is $96$ , as $96 = 32 * 3 = 2^{5} * 3$ .

For two distinct primes $p$ and $q$ let $M (p, q, N)$ be the largest positive integer $\leq N$ only divisible by both $p$ and $q$ and $M (p, q, N) = 0$ if such a positive integer does not exist.

E.g. $M (2, 3, 100) = 96$ .

$M (3, 5, 100) = 75$ and not $90$ because $90$ is divisible by $2, 3$ and $5$ .

Also $M (2, 73, 100) = 0$ because there does not exist a positive integer $\leq 100$ that is divisible by both $2$ and $73$ .

Let $S (N)$ be the sum of all distinct $M (p, q, N)$ . $S (100) = 2262.$

Find $S (10000000)$ .

解决方案

根据分解质因数的思想可以知道，如果 $p, q$ 有其中一个不相同，那么 $M (p, q, N)$ 也不会相同。

因此，令 $N = 10000000$ ，枚举所有的 $p, q (p < q)$ 使得 $p q \leq N$ 。那么找到一对 $(x, y)$ 使得 $p^{x} q^{y} \leq N$ 并且 $p^{x} q^{y}$ 最大，那么 $p^{x} q^{y}$ 便是其中一个答案。

另外一个地方则是，质数只需要筛选到 $\frac{N}{2}$ 即可。因为枚举的 $(p, q)$ 中， $p$ 至少为 $2$ ，那么 $q$ 最多为 $\frac{N}{2}$ 。

代码

from tools import get_prime

N = 10 ** 7
pr = get_prime(N // 2)
ans = 0
for i in range(len(pr) - 1):
    for j in range(i + 1, len(pr)):
        if pr[i] * pr[j] > N:
            break
        mx = 0
        x = pr[i]
        while x * pr[j] <= N:
            y = pr[j]
            while x * y <= N:
                mx = max(mx, x * y)
                y *= pr[j]
            x *= pr[i]
        ans += mx
print(ans)