【BZOJ】【2440】【中山市选2011】完全平方数

莫比乌斯函数/容斥原理

　　PoPoQQQ讲义引入例题= =

　　比较水……就是莫比乌斯函数的简单应用，也可理解为乱容斥一下……

　　二分答案——>求1~x有多少个无平方因子的数Q(x)。

引用一下PoPoQQQ的题解：

•根据容斥原理可知 对于sqrt(x)以内所有的质数 有

•  x以内的无平方因子数

•=0个质数乘积的平方的倍数的数的数量(1的倍数)

•-每个质数的平方的倍数的数的数量(9的倍数,25的倍数,...)

•+每2个质数乘积的平方的倍数的数的数量(36的倍数,100的倍数,...)-...

•容易发现每个乘积a前面的符号恰好是aaarticlea/png;base64,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" alt="" width="40" height="25" />(例如aaarticlea/png;base64,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" alt="" width="74" height="23" />故9对答案的贡献为负；aaarticlea/png;base64,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" alt="" width="55" height="21" />，故36对答案的贡献为正)

•x以内i^2的倍数有aaarticlea/png;base64,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" alt="" width="46" height="61" />个 故有aaarticlea/png;base64,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" alt="" width="136" height="53" />

 /**************************************************************
     Problem: 2440
     User: Tunix
     Language: C++
     Result: Accepted
     Time:2360 ms
     Memory:10060 kb
 ****************************************************************/

 //BZOJ 2440
 #include<cstdio>
 #include<cstdlib>
 #include<cstring>
 #include<iostream>
 #include<algorithm>
 #define rep(i,n) for(int i=0;i<n;++i)
 #define F(i,j,n) for(int i=j;i<=n;++i)
 #define D(i,j,n) for(int i=j;i>=n;--i)
 using namespace std;

 int getint(){
     int v=,sign=; char ch=getchar();
     while(ch<''||ch>'') {if (ch=='-') sign=-; ch=getchar();}
     while(ch>=''&&ch<='') {v=v*+ch-''; ch=getchar();}
     return v*=sign;
 }
 /*******************tamplate********************/
 const int N=1e6+;
 typedef long long LL;
 int mu[N],prime[N];
 bool check[N];
 void getmu(int n){
     int tot=;
     mu[]=;
     for(int i=;i<n;++i){
         if (!check[i]){
             prime[tot++]=i;
             mu[i]=-;
         }
         rep(j,tot){
             if (i*prime[j]>n) break;
             check[i*prime[j]]=;
             if (i%prime[j]) mu[i*prime[j]]=-mu[i];
             else{
                 mu[i*prime[j]]=;
                 break;
             }
         }
     }
 }
 int judge(int x){
     int ans=;
     for(int i=;i*i<=x;i++)
         ans+=mu[i]*((LL)x/(i*i));
     return ans;
 }
 int main(){
     int T=getint(),n;
     getmu(N-);
     while(T--){
         n=getint();
         int l=,r=n*,mid=,now=;
         while(l<r){
             mid=l+(r-l>>);
             now=judge(mid);
             if (now<n) l=mid+;
             else r=mid;
         }
         printf("%d\n",l);
     }
     return ;
 }