洛谷1390 公约数的和

$题意:$

$$求\sum _{i=1}^n\sum _{j=i+1}^n\gcd (i,j)$$

$题解:$

$\begin{array}{rl}原式& =\sum _{i=1}^n\sum _{j=i+1}^n\gcd (i,j)\\ & =\sum _{d=1}^{n}d\cdot \sum _{i=1}^n\sum _{j=i+1}^n[gcd(i,j)=d]\\ & =\sum _{d=1}^{n}d\cdot \sum _{i=1}^{n/d}\sum _{j=i+1}^{n/d}[gcd(i,j)=1]\\ & =\sum _{d=1}^{n}d\cdot \sum _{j=1}^{n/d}\varphi (j)\end{array}$

$因此预处理出欧拉函数的前缀和，然后枚举d即可$

#include 
using namespace std;

typedef long long LL;

const int N = 2000020;
int n, phi[N], p[N];
LL sum[N];
int tot;
bool used[N];

//预处理出素数 欧拉函数 欧拉函数前缀和
void init(LL n) {
	phi[1] = 0;
	for (int i = 2; i <= n; i++) {
		if (!used[i]) {
			phi[i] = i - 1;
			p[++tot] = i;
		}
		for (int j = 1; p[j] * i <= n; j++) {
			used[p[j] * i] = true;
			if (i % p[j] == 0) {
				phi[i * p[j]] = phi[i] * p[j];
				break;
			}
			phi[i * p[j]] = phi[i] * (p[j] - 1);
		}
	}
	for (int i = 1; i <= n; i++) sum[i] = sum[i-1] + phi[i];
}

LL cal(LL x) {
	LL res = 0;
	for (int i = 1; i <= x; i++) {
		int t = x / i;
		res += 1LL * sum[t] * i;
	}
	return res;
}


int main() {
	int x;
	cin >> x;
	init(x);
	cout << cal(x) << endl;
	return 0;
}