Luogu P1829 [国家集训队]Crash的数字表格

题目链接: https://www.luogu.org/problemnew/show/P1829

显然,题目所求为以下式子

以下均默认$n \leq m$

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

先来一些比较显然的东西
$$
\begin{aligned}
\sum_{i = 1}^n \sum_{j = 1}^m lcm(i,j) & =
\sum_{i = 1}^n \sum_{j = 1}^n \frac{i \cdot j}{\gcd(i,j)}\\
& = \sum_{d = 1}^n d \sum_{i = 1}^{\frac{n}{d}} \sum_{j = 1}^{\frac{m}{d}} i \cdot j \cdot [\gcd(i,j) = 1]\\
\end{aligned}
$$
后面一段看起来还可以再加优化


$$
\begin{aligned}
g(n,m)
& = \sum_{i = 1}^n \sum_{j = 1}^m i \cdot j \cdot [\gcd(i,j) = 1] \\
& = \sum_{d = 1}^n d^2 \mu(d) \sum_{i = 1}^{\frac{n}{d}} \sum_{j = 1}^{\frac{m}{d}} i \cdot j \\
\end{aligned}
$$
啥都不说了,除法分块

则答案为
$$
\begin{aligned}
\sum_{i = 1}^n \sum_{j = 1}^m lcm(i,j) & = \sum_{d = 1}^n d \sum_{i = 1}^{\frac{n}{d}} \sum_{j = 1}^{\frac{m}{d}} i \cdot j \cdot [\gcd(i,j) = 1]\\
& = \sum_{d = 1}^n d \cdot g(\frac{n}{d}, \frac{m}{d})
\end{aligned}
$$
又是一个除法分块

两个套在一块,据说积分可得$O(n + m)$,反正我是不会证明

代码

#include <cstdio>

inline int Min(int a, int b){return a < b? a: b;}

const int N = 1e7 + 1e5;
const int mod = 20101009;

int x, y, inv2, pcnt;
int pri[N], mu[N];
bool vis[N];

void pre(){
    inv2 = (mod + 1LL) / 2LL;
    mu[1] = 1; vis[1] = 1;
    for(int i = 2; i < N; i++){
        if(vis[i] == false){
            pri[ ++pcnt ] = i;
            mu[i] = -1;
        }
        for(int j = 1, tmp; j <= pcnt; j++){
            tmp = pri[j] * i;
            if(tmp > N)
                break;
            vis[tmp] = true;
            if(i % pri[j] == 0){
                mu[tmp] = 0;
                break;
            }
            mu[tmp] = -mu[i];
        }
        mu[i] = (mu[i - 1] + ( ( (1LL * i * i) % mod ) * mu[i]) % mod ) % mod;
    }
}

inline int sum(int left, int rig){
    return (((1LL * (rig - left + 1) * (left + rig)) % mod) * inv2) % mod;
}

int g(int n, int m){
    if(n > m){ int tmp = n; n = m; m = tmp; }
    int res = 0;
    for(int left = 1, rig; left <= n; left = rig + 1){
        rig = Min(n / (n / left), m / (m / left));     
        res = (res + ( ( (1LL * (mu[rig] - mu[left - 1]) * sum(1, n / left) ) % mod) * sum(1, m / left) ) % mod) % mod;
    }
    return res;
} 

int calc(int n, int m){
    if(n > m){ int tmp = n; n = m; m = tmp; }
    int res = 0;
    for(int left = 1, rig; left <= n; left = rig + 1){
        rig = Min(n / (n / left), m / (m / left));
        res = (res + (1LL * sum(left, rig) * g(n/left, m/left)) % mod) % mod;
    }        
    return res;
}

int main(){
#ifdef woshiluo
    freopen("luogu.1829.in", "r", stdin);
    freopen("luogu.1829.out", "w", stdout);
#endif
    pre();
    scanf("%d%d", &x, &y);
    printf("%d\n", ( (calc(x, y) % mod) + mod) % mod);
}

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