[Bzoj 2956] 模积和 (整除分块)

整除分块

　一般形式：$\sum_{i = 1}^n \lfloor \frac{n}{i} \rfloor * f(i)$。

　需要一种高效求得函数 $f(i)$ 的前缀和的方法，比如等差等比数列求和或对于积性函数的筛法等，然后就可以用整除分块的思想做。

　

题目解法

　化公式变成比较方便的形式：

　　$\ \sum_{i = 1}^n \sum_{j = 1}^m (n \mod i)(m \mod j), i \ne j$

　$= \sum_{i = 1}^n \sum_{j = 1}^m (n - i \lfloor \frac{n}{i} \rfloor)(m - j \lfloor \frac{m}{j} \rfloor) - \sum_{i = 1}^{min(n, m)} (n - i \lfloor \frac{n}{i} \rfloor)(m - i \lfloor \frac{m}{i} \rfloor)$

　

　乘法分配律展开，化简，令 $t = min(n, m)$ 得：

　　$\ \sum_{i = 1}^n \sum_{j = 1}^m (n - i \lfloor \frac{n}{i} \rfloor)(m - j \lfloor \frac{m}{j} \rfloor) - \sum_{i = 1}^t (n - i \lfloor \frac{n}{i} \rfloor)(m - i \lfloor \frac{m}{i} \rfloor)$

　$= \sum_{i = 1}^n \sum_{i = 1}^m nm + m\sum_{i = 1}^n \sum_{i = 1}^m j \lfloor \frac{n}{i} \rfloor + n\sum_{i = 1}^n \sum_{i = 1}^m \lfloor \frac{m}{j} \rfloor +$

　$\sum_{i = 1}^n \sum_{i = 1}^m ij \lfloor \frac{n}{i} \rfloor \lfloor \frac{m}{j} \rfloor - \sum_{i = 1}^t nm + m \sum_{i = 1}^t i \lfloor \frac{n}{i} \rfloor - n \sum_{i = 1}^t i \lfloor \frac{m}{i} \rfloor - \sum_{i = 1}^t i^2 \lfloor \frac{n}{i} \rfloor$

　$\sum_{i = 1}^n \sum_{i = 1}^m ij \lfloor \frac{n}{i} \rfloor \lfloor \frac{m}{j} \rfloor$ 等于 $\sum_{i = 1}^n i \lfloor \frac{n}{i} \rfloor * \sum_{i = 1}^m j \lfloor \frac{m}{j} \rfloor$，一个一个算即可。

　

　代码写的比较长……因为用 $unsigned ll$ 为了避免出负数也多了很多取模……

#include <queue>

#include <cstdio>

#include <cctype>

#include <cstring>

#include <iostream>

#include <algorithm>

using namespace std;

typedef unsigned long long u64;

const u64 mod = 19940417;

const u64 inv_6 = 3323403;

inline u64 Calc_1(u64 l, u64 r) { return (l + r) * (r - l + 1) / 2 % mod; }

inline u64 Calc_2(u64 x) { return ((x + 1) * (2 * x + 1) % mod) * (x * inv_6 % mod) % mod; }

int main(int argc, const char *argv[])

{

  u64 n = 0, m = 0, t = 0, ans = 0, sum_1 = 0, sum_2 = 0;

  scanf("%llu%llu", &n, &m);

  ans = ((n * m % mod) * (n * m % mod) % mod);

  t = min(n, m), ans = (ans + mod - (n * m % mod) * t % mod) % mod;

  for(u64 tmp, l = 1, r = 1; l <= n; l = r + 1) {

    tmp = n / l, r = n / tmp;

    ans = (ans + mod - (Calc_1(l, r) * tmp % mod) * (m * m % mod) % mod) % mod;

    sum_1 = (sum_1 + Calc_1(l, r) * tmp % mod) % mod;

  }

  for(u64 tmp, l = 1, r = 1; l <= m; l = r + 1) {

    tmp = m / l, r = m / tmp;

    ans = (ans + mod - (Calc_1(l, r) * tmp % mod) * (n * n % mod) % mod) % mod;

    sum_2 = (sum_2 + Calc_1(l, r) * tmp % mod) % mod;

  }

  for(u64 tmp, l = 1, r = 1; l <= t; l = r + 1) {

    tmp = n / l, r = min(t, n / tmp);

    ans = (ans + Calc_1(l, r) * (tmp * m % mod)) % mod;

  }

  for(u64 tmp, l = 1, r = 1; l <= t; l = r + 1) {

    tmp = m / l, r = min(t, m / tmp);

    ans = (ans + Calc_1(l, r) * (tmp * n % mod)) % mod;

  }

  for(u64 l = 1, r = 1; l <= t; l = r + 1) {

    r = min(t, min(n / (n / l), m / (m / l)));

    ans = (ans + mod - (mod + Calc_2(r) - Calc_2(l - 1)) * ((n / l) * (m / l) % mod) % mod) % mod;

  }

  printf("%llu\n", (ans + sum_1 * sum_2) % mod);

  return 0;

}