51nod 1056

n<=10000000000

然后欧拉函数的前缀和可以用莫比乌斯函数的前缀和快速求，注意各种取模

#include<cstdio>
typedef long long i64;
const int N=,P=,M=,I2=(M+)/;
int ps[N/],pp=;
bool isnp[N+];
int ms[N+],phs[N+];
i64 xs[P];
int ys[P];
int f(i64 n){
    if(n<=N)return ms[n];
    int w=n%P;
    while(xs[w]){
        if(xs[w]==n)return ys[w];
        w+=;
        if(w>=P)w-=P;
    }
    xs[w]=n;
    i64 s=;
    for(i64 i=,j;i<=n;i=j+){
        j=n/(n/i);
        s+=(j-i+)*f(n/i);
    }
    return ys[w]=-s;
}
int sum_phi(i64 n){
    if(n<=N)return phs[n];
    i64 ans=;
    for(i64 i=,j;i<=n;i=j+){
        i64 k=n/i;
        j=n/k;
        k%=M;
        ans=(ans+k*k%M*(f(j)-f(i-)))%M;
    }
    ans=(ans+)*I2%M;
    return ans;
}
int main(){
    ms[]=phs[]=;
    for(int i=;i<=N;i++){
        if(!isnp[i])ms[ps[pp++]=i]=-,phs[i]=i-;
        for(int j=;j<pp&&i*ps[j]<=N;j++){
            isnp[i*ps[j]]=;
            if(i%ps[j])ms[i*ps[j]]=-ms[i],phs[i*ps[j]]=phs[i]*(ps[j]-);
            else{
                phs[i*ps[j]]=phs[i]*ps[j];
                break;
            }
        }
    }
    for(int i=;i<=N;i++)ms[i]+=ms[i-];
    for(int i=;i<=N;i++)(phs[i]+=phs[i-])%=M;
    i64 n,ans=;
    scanf("%lld",&n);
    for(i64 i=,j=,k;i<=n;i=j+){
        k=n/i;
        j=n/k;
        k%=M;
        ans=(ans+k*(k+)%M*(sum_phi(j)-sum_phi(i-)))%M;
    }
    n%=M;
    ans=(ans-n*(n+)%M*I2)%M;
    printf("%lld",(ans+M)%M);
    return ;
}