牛客小白月赛12：月月给华华出题（欧拉函数）

月月给华华出题

思路

$$\sum _{i=1}^n\frac{i}{gcd(i,n)}$$

$$=\sum _{d\mid n}\sum _{i=1}^n\frac{i}{d}(gcd(i,d)==d)$$

$$=\sum _{d\mid n}\sum _{i=1}^{\frac{n}{d}}i(gcd(i,d)==1)$$

$$=\sum _{d\mid n}\frac{\frac{n}{d}\ast \varphi (\frac{n}{d})}{2}$$

结果就出来了，我们只要仿照埃氏筛法，统计一遍答案就行了。

代码

/*
  Author : lifehappy
*/
#pragma GCC optimize(2)
#pragma GCC optimize(3)
#include 

#define mp make_pair
#define pb push_back
#define endl '\n'
#define mid (l + r >> 1)
#define lson rt << 1, l, mid
#define rson rt << 1 | 1, mid + 1, r
#define ls rt << 1
#define rs rt << 1 | 1

using namespace std;


typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;

const double pi = acos(-1.0);
const double eps = 1e-7;
const int inf = 0x3f3f3f3f;

inline ll read() {
    ll x = 0, f = 1; char c = getchar();
    while(c < '0' || c > '9') {
        if(c == '-') f = -1;
        c = getchar();
    }
    while(c >= '0' && c <= '9') {
        x = (x << 1) + (x << 3) + (c ^ 48);
        c = getchar();
    }
    return x * f;
}

const int N = 5e6 + 10;

int phi[N], n;

bool st[N];

vector<int> prime;

ll sum[N], ans[N];

void init() {
    st[0] = st[1] = 1;
    phi[1] = 1, sum[1] = 1;//注意把sum[1]置为1，特判。
    for(int i = 2; i < N; i++) {
        if(!st[i]) {
            prime.pb(i);
            phi[i] = i - 1;
            sum[i] = 1ll * i * phi[i] / 2;
        }
        for(int j = 0; j < prime.size() && i * prime[j] < N; j++) {
            st[i * prime[j]] = 1;
            if(i % prime[j]) {
                phi[i * prime[j]] = phi[i] * (prime[j] - 1);
                sum[i * prime[j]] = 1ll * phi[i * prime[j]] * (i * prime[j]) / 2;
            }
            else {
                phi[i * prime[j]] = phi[i] * prime[j];
                sum[i * prime[j]] = 1ll * phi[i * prime[j]] * (i * prime[j]) / 2;
                break;
            }
        }
    }
    for(int i = 1; i < N; i++) {
        for(int j = i; j < N; j += i) {
            ans[j] += sum[i];
        }
    }
}

int main() {
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
    init();
    n = read();
    for(int i = 1; i <= n; i++)
        cout << ans[i] << "\n";
	return 0;
}