[Bzoj 2956] 模积和 (整除分块)

Posted 2021-02-03 nanjoqin

整除分块

　一般形式：(sum_{i = 1}^n lfloor frac{n}{i} floor * 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 e j)
　(= sum_{i = 1}^n sum_{j = 1}^m (n - i lfloor frac{n}{i} floor)(m - j lfloor frac{m}{j} floor) - sum_{i = 1}^{min(n, m)} (n - i lfloor frac{n}{i} floor)(m - i lfloor frac{m}{i} floor))
　
　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} floor)(m - j lfloor frac{m}{j} floor) - sum_{i = 1}^t (n - i lfloor frac{n}{i} floor)(m - i lfloor frac{m}{i} floor))
　(= sum_{i = 1}^n sum_{i = 1}^m nm + msum_{i = 1}^n sum_{i = 1}^m j lfloor frac{n}{i} floor + nsum_{i = 1}^n sum_{i = 1}^m lfloor frac{m}{j} floor +)
　(sum_{i = 1}^n sum_{i = 1}^m ij lfloor frac{n}{i} floor lfloor frac{m}{j} floor - sum_{i = 1}^t nm + m sum_{i = 1}^t i lfloor frac{n}{i} floor - n sum_{i = 1}^t i lfloor frac{m}{i} floor - sum_{i = 1}^t i^2 lfloor frac{n}{i} floor)
　(sum_{i = 1}^n sum_{i = 1}^m ij lfloor frac{n}{i} floor lfloor frac{m}{j} floor) 等于 (sum_{i = 1}^n i lfloor frac{n}{i} floor * sum_{i = 1}^m j lfloor frac{m}{j} floor)，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
", (ans + sum_1 * sum_2) % mod);

  return 0;
}