Farey Sequence （欧拉函数+前缀和）

题目链接：http://poj.org/problem?id=2478

Description

The Farey Sequence Fn for any integer n with n >= 2 is the set of irreducible rational numbers a/b with 0 < a < b <= n and gcd(a,b) = 1 arranged in increasing order. The first few are
F2 = {1/2}
F3 = {1/3, 1/2, 2/3}
F4 = {1/4, 1/3, 1/2, 2/3, 3/4}
F5 = {1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5}

You task is to calculate the number of terms in the Farey sequence Fn.

Input

There are several test cases. Each test case has only one line, which contains a positive integer n (2 <= n <= 10⁶). There are no blank lines between cases. A line with a single 0 terminates the input.

Output

For each test case, you should output one line, which contains N(n) ---- the number of terms in the Farey sequence Fn.

Sample Input

2
3
4
5
0

Sample Output

1
3
5
9

思路：由a / b,gcd(a,b)=1知，当前f(n)是在f（n - 1）的基础上加上以n为分母，与n互质的数为分子的分数，所以f（n）比f（n - 1）增加了1~n内与n互质的数的个数，现在题意就很明显了，就是要求欧拉函数，不过需要一个前缀和。
代码实现如下：

#include <cstdio>
#include <cstring>
#include <iostream>
using namespace std;

#define debug(x) cout <<'[' <<x <<']' <<endl;
const int maxn = 1e6 + 7;
int n, m;
int v[maxn], p[maxn], phi[maxn];
long long sum[maxn];

void euler() {
    m = 0;
    memset(v, 0, sizeof(v));
    memset(sum, 0, sizeof(sum));
    for(int i = 2; i < maxn; i++) {
        if(v[i] == 0) {
            v[i] = i;
            p[m++] = i;
            phi[i] = i - 1;
        }
        for(int j = 0; j < m; j++) {
            if(p[j] > v[i] || p[j] > maxn / i) break;
            v[i * p[j]] = p[j];
            phi[i * p[j]] = phi[i] * (i % p[j] ? p[j] - 1 : p[j]);
        }
    }
    phi[0] = phi[1] = 0;
    for(int i = 1; i < maxn; i++) {
        sum[i] = sum[i - 1] + phi[i];
    }
}

int main() {
    euler();
    while(~scanf("%d", &n) && n) {
//        debug(phi[n]);
        printf("%lld
", sum[n]);
    }
}