• [SDOI2015] 约数个数和


    Description

    (d(x))(x) 的约数个数,给定 (n,m),求 (sum_{i=1}^nsum_{j=1}^md(ij))(n,m,T le 5 imes 10^4)

    Solution

    重要结论如黑色式子所示,其余部分为套路性的反演推导


    #include <bits/stdc++.h>
    using namespace std;
    
    #define int long long
    const int N = 50005;
    const int MAXN = 50005;
    
    bool isNotPrime[MAXN + 1];
    int mu[MAXN + 1], phi[MAXN + 1], primes[MAXN + 1], cnt, h[MAXN+1];
    inline void euler() {
        isNotPrime[0] = isNotPrime[1] = true;
        mu[1] = 1;
        phi[1] = 1;
        for (int i = 2; i <= MAXN; i++) {
            if (!isNotPrime[i]) {
                primes[++cnt] = i;
                mu[i] = -1;
                phi[i] = i - 1;
            }
            for (int j = 1; j <= cnt; j++) {
                int t = i * primes[j];
                if (t > MAXN) break;
                isNotPrime[t] = true;
                if (i % primes[j] == 0) {
                    mu[t] = 0;
                    phi[t] = phi[i] * primes[j];
                    break;
                } else {
                    mu[t] = -mu[i];
                    phi[t] = phi[i] * (primes[j] - 1);
                }
            }
        }
        for(int i=1;i<=MAXN;i++) mu[i]+=mu[i-1];
    }
    
    int calch(signed x) {
        signed l=1;
        int ans=0;
        while(l<=x) {
            signed r=x/(x/l);
            ans+=1ll*(r-l+1)*(x/l);
            l=r+1;
        }
        return ans;
    }
    
    int solve(int n,int m) {
        if(n==0 || m==0) return 0;
        int ans=0,l=1,r=0;
        if(n>m) swap(n,m);
        while(l<=n) {
            r=min(n/(n/l),m/(m/l));
            ans+=(mu[r]-mu[l-1])*h[n/l]*h[m/l];
            l=r+1;
        }
        return ans;
    }
    
    signed main() {
        euler();
        int t,a,b,c,d,k;
        ios::sync_with_stdio(false);
        cin>>t;
        for(int i=1;i<=5e4;i++) h[i]=calch(i);
        while(t--) {
            int n,m;
            cin>>n>>m;
            cout<<solve(n,m)<<endl;
        }
    }
    
    
  • 相关阅读:
    java.lang.NoSuchMethodError: org.springframework.web.context.request.ServletRequestAttributes.<init>
    eclipse web项目实际工程路径对应
    java中专业术语详解
    Maven详解
    工作常用
    html页面布局
    jQuery易混淆概念的区别
    Jquery Datagrid
    Jquery EasyUI 动态添加标签页(Tabs)
    sql语句的写法
  • 原文地址:https://www.cnblogs.com/mollnn/p/13155293.html
Copyright © 2020-2023  润新知