2019南昌网络邀请赛 G(莫比乌斯反演)

G. tsy's number

#include<bits/stdc++.h>
#define me(a,x) memset(a,x,sizeof(a))
#define IN freopen("in.txt","r",stdin);
#define OUT freopen("out.txt","w",stdout);
#define sc scanf
#define itn int
#define STR clock_t startTime = clock();
#define END clock_t endTime = clock();cout << double(endTime - startTime) / CLOCKS_PER_SEC *1000<< "ms" << endl;
using namespace std;
const int N=1e7+5;
const long long mod=1LL<<30;
const long long mod2=998244353;
const int oo=0x7fffffff;
const int sup=0x80000000;
typedef long long ll;
typedef unsigned long long ull;
template <typename it>void db(it *begin,it *end){while(begin!=end)cout<<(*begin++)<<" ";puts("");}
template <typename it>
string to_str(it n){string s="";while(n)s+=n%10+'0',n/=10;reverse(s.begin(),s.end());return s;}
template <typename it>int o(it a){cout<<a<<endl;return 0;}
inline ll mul(ll a,ll b,ll c){ll ans=0;for(;b;b>>=1,a=(a+a)%c)if(b&1)ans=(ans+a)%c;return ans;}
inline ll ksm(ll a,ll b,ll c){ll ans=1;for(;b;b>>=1,a=mul(a,a,c))if(b&1)ans=mul(ans,a,c);return ans;}
inline void exgcd(ll a,ll b,ll &x,ll &y){if(!b)x=1,y=0;else exgcd(b,a%b,y,x),y-=x*(a/b);}
ll inv_1(ll x){return x;}
ll inv_2(ll x){return x*(x+1)%(2LL*mod)/2;}
ll inv_6(ll x){return x*(x+1)%(6LL*mod)*(2LL*x+1)%(6LL*mod)/6;}
int phi[N],prime[N],tot=0;
short int mu[N];
bool vis[N]={0};
ll F[N];
void f_mod(ll &x){
    if(x>=mod)x-=x/mod*mod;
    if(x<0)x+=mod;
}
void pre(){
    F[1]=mu[1]=phi[1]=vis[1]=1;
    for(int i=2;i<N;i++){
        if(!vis[i])prime[++tot]=i,phi[i]=i-1,mu[i]=-1,F[i]=i-2;
        for(int j=1;j<=tot&&i*prime[j]<N;j++){
            vis[i*prime[j]]=1;
            if(i%prime[j]==0){
                mu[i*prime[j]]=0;
                phi[i*prime[j]]=phi[i]*prime[j];
                if(i==prime[j])F[i*prime[j]]=1LL*(prime[j]-1)*(prime[j]-1)%mod;
                else if(i%(1LL*prime[j]*prime[j])==0)F[i*prime[j]]=F[i]*prime[j]%mod;
                else F[i*prime[j]]=F[i/prime[j]]*(1LL*(prime[j]-1)*(prime[j]-1)%mod)%mod;
                break;
            }else mu[i*prime[j]]=-mu[i],phi[i*prime[j]]=phi[i]*(prime[j]-1),F[i*prime[j]]=F[i]*F[prime[j]]%mod;
        }
    }
    for(int i=1;i<N;i++)F[i]=(1LL*i*i%mod*i%mod*F[i]%mod+F[i-1])%mod;
}
int main(){
   // STR
    pre();
    int t;cin>>t;
    while(t--){
        int n;sc("%d",&n);
        ll ans=0;   
        for(int i=1,last;i<=n;i=last+1){
            last=n/(n/i);
            ll x=inv_1(n/i)*inv_2(n/i);
            f_mod(x);
            x*=inv_6(n/i);
            f_mod(x);
            x*=F[last]-F[i-1]+mod;
            f_mod(x);
            ans+=x;
            f_mod(ans);
        }
        printf("%lld\n",ans);
    }
   // END
}