【POJ 2154】 Color （置换、burnside引理）

Color

Description

Beads of N colors are connected together into a circular necklace of N beads (N<=1000000000). Your job is to calculate how many different kinds of the necklace can be produced. You should know that the necklace might not use up all the N colors, and the repetitions that are produced by rotation around the center of the circular necklace are all neglected.

You only need to output the answer module a given number P.

Input

The first line of the input is an integer X (X <= 3500) representing the number of test cases. The following X lines each contains two numbers N and P (1 <= N <= 1000000000, 1 <= P <= 30000), representing a test case.

Output

For each test case, output one line containing the answer.

Sample Input

5
1 30000
2 30000
3 30000
4 30000
5 30000

Sample Output

1
3
11
70
629

Source

【分析】　　

　　经典的置换和burnside引理应用。只有旋转。

　　考虑旋转i个单位，那么循环节有gcd(n,i)个

　　可以化出求 sigma(n^gcd(i,n))/n

　　根据莫比乌斯反演得 sigma(n^d*phi[n/d])/n   (d|n) 【表示我莫比乌斯反演白学了ORZ

　　即sigma(n^(d-1)*phi[n/d]) (d|n)

　　就算一算就好了。【不用欧拉筛，筛不到n的，先求出n的质因子，那么d的质因子也在里面，然后根据定义分解质因数求phi

　　最后要除以n，不能求逆元哦，每次少乘一个n就好了。

　　然而我还WA着！！！！【AC了再放代码吧

　　好了AC了：

 #include<cstdio>
 #include<cstdlib>
 #include<cstring>
 #include<iostream>
 #include<algorithm>
 using namespace std;
 #define Maxn 35000 int n,p;
 int pri[Maxn],phi[Maxn],pl;
 int d[Maxn],dl;
 bool vis[Maxn]; void init()
 {
     pl=;
     memset(vis,,sizeof(vis));
     for(int i=;i<=Maxn-;i++)
     {
         if(vis[i]==)
         {
             pri[++pl]=i;
             phi[i]=i-;
         }
         for(int j=;j<=pl;j++)
         {
             if(i*pri[j]>Maxn-) break;
             vis[i*pri[j]]=;
             if(i%pri[j]!=) phi[i*pri[j]]=phi[i]*(pri[j]-);
             else phi[i*pri[j]]=phi[i]*pri[j];
             if(i%pri[j]==) break;
         }
     }
     phi[]=;
     // for(int i=2;i<=10;i++) printf("%d ",phi[i]);
     // printf("\n");
 } int qpow(int a,int b,int p)
 {
     a%=p;
     int ans=;
     while(b)
     {
         if(b&) ans=(ans*a)%p;
         a=(a*a)%p;
         b>>=;
     }
     return ans;
 } void get_d(int n)
 {
     dl=;
     for(int i=;i<=pl;i++) if(n%pri[i]==)
     {
         while(n%pri[i]==) n/=pri[i];
         d[++dl]=pri[i];
     }
     if(n!=) d[++dl]=n;
 } int gphi(int x)
 {
     int ans=x;
     for(int i=;i<=dl;i++) if(x%d[i]==)
     {
         ans=ans/d[i]*(d[i]-);
     }
     return ans;
 } int main()
 {
     int T;
     scanf("%d",&T);
     init();
     while(T--)
     {
         scanf("%d%d",&n,&p);
         get_d(n);
         int ans=;
         for(int i=;i*i<=n;i++) if(n%i==)
         {
             ans=(ans+(qpow(n,i-,p)*(gphi(n/i)%p)))%p;
             if(i*i!=n) ans=(ans+(qpow(n,n/i-,p)*(phi[i]%p)))%p;
         }
         printf("%d\n",ans);
     }
     return ;
 }

2017-01-13 11:48:09