poj2409

用n个颜色的珠子编项链，求有多少种情况

由N(G,C) = 所有f的稳定核的和/|G|

m边形有m种旋转m种翻转

首先说旋转，有模线性方程可知每种旋转都有gcd(m,i)个循环节且每个循环节长度为n/gcd(m,i)

所以每个旋转的稳定核 = pow(n,gcd(m,i))

翻转的循环节数可有观察得知

#include<algorithm>
#include<string.h>
#include<stdio.h>
#include<cmath>
#include<iostream>
using namespace std;
#define ll long long
int main(){
    //freopen("in.cpp", "r", stdin);
    ll ans;
    int n, m;
    while(scanf("%d%d", &n, &m), n+m){
        ans = 0;
        ll sum =  1;
        for(int i =1; i <= m; i++){
            sum += pow(n, __gcd(i, m));
        }
        ans = sum;//cout<<"*"<<ans<<endl;
        if(m & 1){
            sum = 1;
            for(int i = 0; i < (m+1)/2; i++){
                sum *= n;
            }
            sum *= m;
            ans += sum;
        }else{
            sum = 1;
            for(int i = 0; i < m/2; i++){
                sum *= n;
            }
            ans += m/2*(sum + sum*n);
        }//cout<<ans<<endl;
        ans/= m*2;
        cout<<ans<<endl;
    }
}