HDU 2841 Visible Trees（数论）

标题效果：给你个m*n方格，广场格从（1，1）开始。

在树中的每个点，然后让你(0,0)点往下看，问：你能看到几棵树。

解题思路：假设你的视线被后面的树和挡住的话以后在这条线上的树你是都看不见的啊。挡住的话就是这个小的方格内对角线的连线过顶点，如图：

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建设A为(0,0)假设B能遮挡住C，那么B与C组成的矩形中nx, mx是不互质的。

所以x从1開始枚举n假设m中某一项与x互质那么就会有一个格子显示出来，所以这里须要找的是1-n中与枚举的x互质的个数。这里的话我们能够用到分解质因数，然后用状态压缩进行容斥，求出n中与x互质的个数。

PS：我代码里面写的n，m与这里解释的n，m是反过来的。

Visible Trees

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)

Total Submission(s): 1556    Accepted Submission(s): 645

Problem Description

There are many trees forming a m * n grid, the grid starts from (1,1). Farmer Sherlock is standing at (0,0) point. He wonders how many trees he can see.



If two trees and Sherlock are in one line, Farmer Sherlock can only see the tree nearest to him.

 

Input

The first line contains one integer t, represents the number of test cases. Then there are multiple test cases. For each test case there is one line containing two integers m and n(1 ≤ m, n ≤ 100000)

 

Output

For each test case output one line represents the number of trees Farmer Sherlock can see.

 

Sample Input

2
1 1
2 3

 

Sample Output

1
5

#include <algorithm>
#include <iostream>
#include <stdlib.h>
#include <string.h>
#include <iomanip>
#include <stdio.h>
#include <string>
#include <queue>
#include <cmath>
#include <stack>
#include <map>
#include <set>
#define eps 1e-8
#define M 1000100
#define LL long long
//#define LL long long
#define INF 0x3f3f3f
#define PI 3.1415926535898
#define mod 1000000007

const int maxn = 100010;

using namespace std;

struct node
{
    int num[8];
    int ans;
} p[maxn];

int f[maxn];
int k[maxn];
int t;
void prime()
{
    t = 0;
    memset(f, 0, sizeof(f));
    for(int i = 2; i <= maxn-5; i++)
    {
        if(!f[i]) k[t++] = i;
        for(int j = 0; j < t; j++)
        {
            if(i*k[j] > maxn-5) break;
            f[i*k[j]] = true;
            if(i%k[j] == 0) break;
        }
    }
}

LL judge(int n, int x)
{
    LL cnt = 0;
    for(int i = 0; i < (1<<(p[x].ans)); i++)
    {
        int ss = 0;
        int sx = 1;
        for(int j = 0; j < p[x].ans; j++)
        {
            if((1<<j)&i)
            {
                ss++;
                sx *= p[x].num[j];
            }
        }
        if(ss%2) cnt += (n-n/sx);
        else cnt -= (n-n/sx);
    }
    return cnt;
}

int main()
{
    prime();
    for(int i = 2; i <= maxn-10; i++)
    {
        int x = i;
        p[i].ans = 0;
        for(int j = 0; j < t; j++)
        {
            if(x == 1) break;
            if(!f[x])
            {
                p[i].num[p[i].ans++] = x;
                break;
            }
            if(x%k[j]) continue;
            p[i].num[p[i].ans++] = k[j];
            while(x%k[j] == 0) x /= k[j];
        }
    }
    int n, m;
    int T;
    cin >>T;
    while(T--)
    {
        scanf("%d %d",&n, &m);
        LL sum = n;
        for(int i = 2; i <= m; i++) sum += judge(n, i);
        cout<<sum<<endl;
    }
    return 0;
}

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