线性筛-euler，强大O(n)

Posted 2020-09-03 Jason-Cow

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]==0){phi[i*p[j]]=phi[i]*p[j];break;}
phi[i*p[j]]=phi[i]*(p[j]-1);

一个练手题Hdu1286

 1 #include <algorithm>
 2 #include <iostream>
 3 #include <cstring>
 4 #include <cstdlib>
 5 #include <cstdio>
 6 #include <vector>
 7 #include <cmath>
 8 #include <queue>
 9 #include <map>
10 #include <set>
11 using namespace std;
12 #define file(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout)
13 inline void read(int &ans){
14   ans=0;char x=getchar();int f=0;
15   while(x<\'0\'||x>\'9\'){if(x==\'-\')f=1;x=getchar();}
16   while(x>=\'0\'&&x<=\'9\')ans=ans*10+x-\'0\',x=getchar();
17   if(f)ans=-ans;
18 }
19 
20 const int maxn=50000+10;
21 int phi[maxn],p[maxn],flag[maxn],cnt;
22 void euler(int n){
23   phi[1]=1;
24   for(int i=2;i<=n;i++){
25     if(!flag[i])p[++cnt]=i,phi[i]=i-1;
26     for(int j=1;j<=cnt && i*p[j]<=n;j++){
27       flag[i*p[j]]=1;
28       if(i%p[j]==0){phi[i*p[j]]=phi[i]*p[j];break;}
29       phi[i*p[j]]=phi[i]*(p[j]-1);
30     }
31   }
32 }
33 
34 int main(){
35   euler(32768);
36   int CN,N;
37   for(read(CN);CN--;)read(N),printf("%d\\n",phi[N]);
38   return 0;
39 }

Hdu1286

Hdu1787线性筛O(n)，MLE，怎么办？在线算

1 int euler(int n){
2   int ans=n;
3   for(int i=2;i*i<=n;i++){
4     if(n%i==0)n/=i,ans-=ans/i;
5     while(n%i==0)n/=i;
6   }
7   if(n>1)ans-=ans/n;
8   return ans;
9 }

euler

AC code:

 

 1 #include <algorithm>
 2 #include <iostream>
 3 #include <cstring>
 4 #include <cstdlib>
 5 #include <cstdio>
 6 #include <vector>
 7 #include <cmath>
 8 #include <queue>
 9 #include <map>
10 #include <set>
11 using namespace std;
12 #define file(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout)
13 inline bool read(int &ans){
14   ans=0;char x=getchar();int f=0;
15   while(x<\'0\'||x>\'9\'){if(x==\'-\')f=1;x=getchar();}
16   while(x>=\'0\'&&x<=\'9\')ans=ans*10+x-\'0\',x=getchar();
17   if(f)ans=-ans;
18   return !!ans;
19 }
20 /*
21 const int maxn=(int)1e8+10;
22 int phi[maxn],p[maxn],flag[maxn],cnt;
23 void euler(int n){
24   phi[1]=1;
25   for(int i=2;i<=n;i++){
26     if(!flag[i])p[++cnt]=i,phi[i]=i-1;
27     for(int j=1;j<=cnt && i*p[j]<=n;j++){
28       flag[i*p[j]]=1;
29       if(i%p[j]==0){phi[i*p[j]]=phi[i]*p[j];break;}
30       phi[i*p[j]]=phi[i]*(p[j]-1);
31     }
32   }
33 }
34 */
35 
36 int euler(int n){
37   int ans=n;
38   for(int i=2;i*i<=n;i++){
39     if(n%i==0)n/=i,ans-=ans/i;
40     while(n%i==0)n/=i;
41   }
42   if(n>1)ans-=ans/n;
43   return ans;
44 }
45 
46 int main(){
47   //euler((int)1e8);cout<<euler(1)<<endl;
48   for(int N;read(N);)printf("%d\\n",N-euler(N)-1);
49   return 0;
50 }

Hdu1787

 

Hdu2824，sigma(a,b) phi(x)

 1 #include <algorithm>
 2 #include <iostream>
 3 #include <cstring>
 4 #include <cstdlib>
 5 #include <cstdio>
 6 #include <vector>
 7 #include <cmath>
 8 #include <queue>
 9 #include <map>
10 #include <set>
11 using namespace std;
12 
13 typedef long long ll;
14 const int maxn=(int)3e6+10;
15 ll phi[maxn];int p[maxn],cnt;bool flag[maxn];
16 void euler(int n){
17   phi[1]=1;
18   for(int i=2;i<=n;i++){
19     if(!flag[i])p[++cnt]=i,phi[i]=i-1;
20     for(int j=1;j<=cnt && i*p[j]<=n;j++){
21       flag[i*p[j]]=1;
22       if(i%p[j]==0){phi[i*p[j]]=phi[i]*p[j];break;}
23       phi[i*p[j]]=phi[i]*(p[j]-1);
24     }
25   }
26   for(int i=1;i<=n;i++)phi[i]+=phi[i-1];
27 }
28 
29 int main(){
30   euler((int)3e6);
31   for(int a,b;scanf("%d%d",&a,&b)==2;)printf("%lld\\n",phi[b]-phi[a-1]);
32   return 0;
33 }

Hdu2824