【HDU 6715】算術（莫比烏斯反演）

傳送門

把$lcm$化掉之後枚舉$gcd$可以得到
$ans=\sum_{d=1}\sum_{i=1}^{\frac n d}\sum_{j=1}^{\frac m d}\mu(ijd)[gcd(i,j)=1]$
由於有一個$gcd(i,j)=1$的限制
可以把$\mu(ijd)$拆成$\mu(id)*\mu(jd)/\mu(d)=\mu(id)\mu(jd)\mu(d)$
$=\sum_{d=1}\mu(d)\sum_{i=1}^{\frac n d}\mu(id)\sum_{j=1}^{\frac m d}\mu(jd)[gcd(i,j)=1]$
$=\sum_{d=1}\mu(d)\sum_{k=1}^{\frac n d}\mu(k)\sum_{i=1}^{\frac n{kd}}\mu(ikd)\sum_{j=1}^{\frac m{kd}}\mu(jkd)$
$=\sum_{T=1}^{\min(n,m)}\sum_{d|T}\mu(d)\mu(\frac T d)\sum_{i=1}^{\frac n T}\mu(iT)\sum_{j=1}^{\frac m T}\mu(jT)$

調和級數預處理
每次調和級數計算即可

#include<bits/stdc++.h>
using namespace std;
#define cs const
#define re register
#define pb push_back
#define pii pair<int,int>
#define ll long long
#define fi first
#define se second
#define bg begin
cs int RLEN=1<<20|1;
inline char gc(){
    static char ibuf[RLEN],*ib,*ob;
    (ib==ob)&&(ob=(ib=ibuf)+fread(ibuf,1,RLEN,stdin));
    return (ib==ob)?EOF:*ib++;
}
inline int read(){
    char ch=gc();
    int res=0;bool f=1;
    while(!isdigit(ch))f^=ch=='-',ch=gc();
    while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
    return f?res:-res;
}
inline ll readll(){
    char ch=gc();
    ll res=0;bool f=1;
    while(!isdigit(ch))f^=ch=='-',ch=gc();
    while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
    return f?res:-res;
}
inline int readstring(char *s){
    int top=0;char ch=gc();
    while(isspace(ch))ch=gc();
    while(!isspace(ch)&&ch!=EOF)s[++top]=ch,ch=gc();
    return top;
}
template<typename tp>inline void chemx(tp &a,tp b){a<b?a=b:0;}
template<typename tp>inline void chemn(tp &a,tp b){a>b?a=b:0;}
cs int N=1000005;
int mu[N],vis[N],pr[N],f[N],tot;
inline void init_sieve(cs int n=N-5){
    mu[1]=1;
    for(int i=2;i<=n;i++){
        if(!vis[i])pr[++tot]=i,mu[i]=-1;
        for(int j=1,p;j<=tot&&i*pr[j]<=n;j++){
            p=i*pr[j],vis[p]=1;
            if(i%pr[j]==0)break;
            mu[p]=-mu[i];
        }
    }
    for(int i=1;i<=n;i++)
    for(int j=1;i*j<=n;j++)
    f[i*j]+=mu[i]*mu[j];
}
int n,m;
int main(){
    #ifdef Stargazer
    freopen("lx.in","r",stdin);
    #endif
    init_sieve();
    int T=read();
    while(T--){
        n=read(),m=read();
        ll res=0;
        if(n>m)swap(n,m);
        for(int T=1,v1,v2,i;T<=n;T++){
            v1=0,v2=0,i=T;
            for(;i<=n;i+=T)v1+=mu[i],v2+=mu[i];
            for(;i<=m;i+=T)v2+=mu[i];
            res+=f[T]*v1*v2;
        }
        cout<<res<<'\n';
    }
}