线性筛-euler，强大O(n)

欧拉函数是少于或等于n的数中与n互质的数的数目

φ(1)=1(定义)

类似与莫比乌斯函数，基于欧拉函数的积性

φ(xy)=φ(x)φ(y)

由唯一分解定理展开显然，得证

精髓在于对于积性的应用：

if(i%p[j]==){phi[i*p[j]]=phi[i]*p[j];break;}
phi[i*p[j]]=phi[i]*(p[j]-);

一个练手题Hdu1286

 #include <algorithm>
 #include <iostream>
 #include <cstring>
 #include <cstdlib>
 #include <cstdio>
 #include <vector>
 #include <cmath>
 #include <queue>
 #include <map>
 #include <set>
 using namespace std;
 #define file(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout)
 inline void read(int &ans){
   ans=;char x=getchar();int f=;
   while(x<''||x>''){if(x=='-')f=;x=getchar();}
   while(x>=''&&x<='')ans=ans*+x-'',x=getchar();
   if(f)ans=-ans;
 }
 
 const int maxn=+;
 int phi[maxn],p[maxn],flag[maxn],cnt;
 void euler(int n){
   phi[]=;
   for(int i=;i<=n;i++){
     if(!flag[i])p[++cnt]=i,phi[i]=i-;
     for(int j=;j<=cnt && i*p[j]<=n;j++){
       flag[i*p[j]]=;
       if(i%p[j]==){phi[i*p[j]]=phi[i]*p[j];break;}
       phi[i*p[j]]=phi[i]*(p[j]-);
     }
   }
 }
 
 int main(){
   euler();
   int CN,N;
   for(read(CN);CN--;)read(N),printf("%d\n",phi[N]);
   return ;
 }

Hdu1286

Hdu1787线性筛O(n)，MLE，怎么办？在线算

 int euler(int n){
   int ans=n;
   for(int i=;i*i<=n;i++){
     if(n%i==)n/=i,ans-=ans/i;
     while(n%i==)n/=i;
   }
   if(n>)ans-=ans/n;
   return ans;
 }

euler

AC code:

 #include <algorithm>
 #include <iostream>
 #include <cstring>
 #include <cstdlib>
 #include <cstdio>
 #include <vector>
 #include <cmath>
 #include <queue>
 #include <map>
 #include <set>
 using namespace std;
 #define file(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout)
 inline bool read(int &ans){
   ans=;char x=getchar();int f=;
   while(x<''||x>''){if(x=='-')f=;x=getchar();}
   while(x>=''&&x<='')ans=ans*+x-'',x=getchar();
   if(f)ans=-ans;
   return !!ans;
 }
 /*
 const int maxn=(int)1e8+10;
 int phi[maxn],p[maxn],flag[maxn],cnt;
 void euler(int n){
   phi[1]=1;
   for(int i=2;i<=n;i++){
     if(!flag[i])p[++cnt]=i,phi[i]=i-1;
     for(int j=1;j<=cnt && i*p[j]<=n;j++){
       flag[i*p[j]]=1;
       if(i%p[j]==0){phi[i*p[j]]=phi[i]*p[j];break;}
       phi[i*p[j]]=phi[i]*(p[j]-1);
     }
   }
 }
 */
 
 int euler(int n){
   int ans=n;
   for(int i=;i*i<=n;i++){
     if(n%i==)n/=i,ans-=ans/i;
     while(n%i==)n/=i;
   }
   if(n>)ans-=ans/n;
   return ans;
 }
 
 int main(){
   //euler((int)1e8);cout<<euler(1)<<endl;
   for(int N;read(N);)printf("%d\n",N-euler(N)-);
   return ;
 }

Hdu1787

Hdu2824，sigma(a,b) phi(x)

 #include <algorithm>
 #include <iostream>
 #include <cstring>
 #include <cstdlib>
 #include <cstdio>
 #include <vector>
 #include <cmath>
 #include <queue>
 #include <map>
 #include <set>
 using namespace std;
 
 typedef long long ll;
 const int maxn=(int)3e6+;
 ll phi[maxn];int p[maxn],cnt;bool flag[maxn];
 void euler(int n){
   phi[]=;
   for(int i=;i<=n;i++){
     if(!flag[i])p[++cnt]=i,phi[i]=i-;
     for(int j=;j<=cnt && i*p[j]<=n;j++){
       flag[i*p[j]]=;
       if(i%p[j]==){phi[i*p[j]]=phi[i]*p[j];break;}
       phi[i*p[j]]=phi[i]*(p[j]-);
     }
   }
   for(int i=;i<=n;i++)phi[i]+=phi[i-];
 }
 
 int main(){
   euler((int)3e6);
   for(int a,b;scanf("%d%d",&a,&b)==;)printf("%lld\n",phi[b]-phi[a-]);
   return ;
 }

Hdu2824