P4213 【模板】杜教筛

[题目链接] https://www.luogu.org/problemnew/show/P4213

给定一个正整数\(N(N\le2^{31}-1)\)

求

\(ans_1=\sum_{i=1}^n\varphi(i)\)

\(ans_2=\sum_{i=1}^n \mu(i)\)

[题解] https://acfcacfca.blog.luogu.org/dls-tql

// luogu-judger-enable-o2
#include<bits/stdc++.h>
#include<tr1/unordered_map>
//#define int long long
using namespace std;
typedef long long LL;
const int INF=1e9+7;
inline LL read(){
    register LL x=0,f=1;register char c=getchar();
    while(c<48||c>57){if(c=='-')f=-1;c=getchar();}
    while(c>=48&&c<=57)x=(x<<3)+(x<<1)+(c&15),c=getchar();
    return f*x;
}

const int MAXN=5e6+5;

LL mu[MAXN];LL phi[MAXN];
int prime[MAXN];bool vis[MAXN];
unordered_map <int,LL> Smu,Sphi;
int n,T;

inline void init(int n){
    mu[1]=1,phi[1]=1;
    for(int i=2;i<=n;i++){
        if(!vis[i]){
            prime[++prime[0]]=i;
            mu[i]=-1;
            phi[i]=i-1;
        }
        int x;
        for(int j=1;j<=prime[0]&&(x=(i*prime[j]))<=n;j++){
            vis[x]=1;
            if(i%prime[j]==0){
                mu[x]=0;
                phi[x]=1ll*phi[i]*prime[j];
                break;
            }
            mu[x]=-mu[i];
            phi[x]=1ll*phi[i]*phi[prime[j]];
        }
    }
    for(int i=2;i<=n;i++)
        mu[i]+=mu[i-1],phi[i]+=phi[i-1];
}

inline LL S_mu(int x){
    if(x<MAXN) return mu[x];
    if(Smu[x]) return Smu[x];
    LL res=1;
    for(int l=2,r;l<=x;l=r+1){
        r=x/(x/l);
        res-=1ll*(r-l+1)*S_mu(x/l);
    }
    return Smu[x]=res;
}

inline LL S_phi(int x){
    if(x<MAXN) return phi[x];
    if(Sphi[x]) return Sphi[x];
    LL res=1LL*x*(x+1)/2;//对int敏感!!
    for(int l=2,r;l<=x;l=r+1){
        r=x/(x/l);
        res-=1LL*(r-l+1)*S_phi(x/l);
    }
    return Sphi[x]=res;
}

signed main(){
    T=read();
    init(MAXN-1);
    while(T--){
        n=read();
        printf("%lld %lld\n",S_phi(n),S_mu(n));
    }
}