BZOJ 2818: Gcd( 欧拉函数 )

要求gcd(x, y) = p (1 <= x, y <= n, p为质数 ) 的数对(x, y)个数.我们枚举素数p, 令x' = x / p, y' = y / p, 则只须求  f(p) = gcd(x', y') = 1的数对(x', y')个数(1 <= x', y' <= n / p), 显然f(p) = (∑ phi(x')) * 2 - 1(1 <= x' <= n / p). 所以最后答案为 ∑f(p) 

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#include<bits/stdc++.h>

 

#define clr(x, c) memset(x, c, sizeof(x))

#define rep(i, n) for(int i = 0; i < n; ++i)

#define foreach(i, x) for(__typeof(x.begin()) i = x.begin(); i != x.end(); i++)

 

using namespace std;

 

typedef long long ll;

 

const int maxn = 10000009;

 

int prime[maxn], N = 0, n;

ll phi[maxn], ans = 0;

bool check[maxn];

 

void init() {

clr(check, 0);

phi[1] = 1;

for(int i = 2; i <= n; i++) {

if(!check[i]) {

prime[N++] = i;

phi[i] = i - 1;

}

for(int j = 0; j < N && prime[j] * i <= n; j++) {

check[prime[j] * i] = true;

if(i % prime[j])

   phi[i * prime[j]] = phi[i] * (prime[j] - 1);

else {

phi[i * prime[j]] = phi[i] * prime[j];

break;

}

}

}

}

 

int main() {

freopen("test.in", "r", stdin);

cin >> n;

init();

for(int i = 2; i <= n; i++)

   phi[i] += phi[i - 1];

rep(i, N) ans += phi[n / prime[i]] * 2 - 1;

cout << ans << " ";

return 0;

}

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2818: Gcd

Time Limit: 10 Sec  Memory Limit: 256 MB
Submit: 2450  Solved: 1086
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Description

给定整数N，求1<=x,y<=N且Gcd(x,y)为素数的
数对(x,y)有多少对.

Input

一个整数N

Output

如题

Sample Input

4

Sample Output

4

HINT

hint

对于样例(2,2),(2,4),(3,3),(4,2)

1<=N<=10^7

Source

湖北省队互测