[数论] Codeforces 1717E Madoka and The Best University

题目大意

求

$$\sum_{a>0,b>0,c>0,a+b+c=n}\mathrm{lcm}(c,\gcd(a,b))$$

$(3\leq n\leq 10^5)$

题解

$$ans=\sum_{a}\sum_{b} \mathrm{lcm}(n-a-b,\gcd(a,b))\\ =\sum_{d=1}^n\sum_{a}\sum_{b}\mathrm{lcm}(n-a-b,d)[\gcd(a,b)=d]\\ =\sum_{d=1}^n\sum_{x=1}^{\lfloor\frac{n-1}{d}\rfloor}\sum_{p=1}^{x-1}\mathrm{lcm}(n-xd,d)[\gcd(p,x-p)=1]\\ =\sum_{d=1}^n\sum_{x=1}^{\lfloor\frac{n-1}{d}\rfloor}\mathrm{lcm}(n-xd,d)\sum_{p=1}^{x-1}[\gcd(p,x-p)=1]$$

令 $f(n)=\sum_{i=1}^{n-1}[\gcd(i,n-i)=1]$，

$$f(n)=\sum_{i=1}^{n-1}[\gcd(i,n-i)=1]\\ =\sum_{i=1}^{n-1}\sum_{c|\gcd(i,n-i)}\mu(c)\\ =\sum_{c=1}^{n-1}\mu(c)\sum_{i=1}^{n-1}[c|\gcd(i,n-i)]\\ =\sum_{c|n}\mu(c)\left(\left\lfloor\frac{n}{c}\right\rfloor-1\right)$$

所以可以在 $O(n\sqrt n)$ 的时间内求出全体 $f(1)\sim f(n)$ 。

$$ans=\sum_{d=1}^n\sum_{x=1}^{\lfloor\frac{n-1}{d}\rfloor}\mathrm{lcm}(n-xd,d)f(x)\\ =\sum_{x=1}^n\sum_{d=1}^{\lfloor\frac{n-1}{x}\rfloor}\mathrm{lcm}(n-xd,d)f(x)\\ =\sum_{x=1}^n f(x)\sum_{d=1}^{\lfloor\frac{n-1}{x}\rfloor}\mathrm{lcm}(n-xd,d)$$

注意到对于每个 $x$，$d$ 只会枚举到 $\lfloor\frac{n-1}{x} \rfloor$，所以该式可以 $O(n\log n)$ 求出。

最终时间复杂度 $O(n\sqrt n + n\log n)$。

Code

#include <bits/stdc++.h>
using namespace std;

#define LL long long

const LL MOD = 1e9 + 7;
const int maxn = 100005;

int mu[maxn];
bool not_prime[maxn];
vector<int> prime;

void get_mu(int len) {
    mu[1] = not_prime[1] = 1;
    for (int i = 2;i <= len;++i) {
        if (!not_prime[i]) { prime.push_back(i); mu[i] = -1; }
        for (auto j : prime) {
            int mid = i * j;
            if (mid > len) break;
            not_prime[mid] = 1;
            if (i % j == 0) { mu[mid] = 0; break; }
            mu[mid] = -mu[i];
        }
    }
}

int f[maxn], g[maxn];

int main() {
    get_mu(100000);
    int n;cin >> n;
    for (int x = 1;x <= n;++x) {
        int c = 1;
        for (c = 1;c * c < x;++c) {
            if (x % c == 0) {
                f[x] += mu[c] * (x / c - 1);
                f[x] += mu[x / c] * (c - 1);
            }
        }
        if (c * c == x) f[x] += mu[c] * (x / c - 1);
    }
    LL ans = 0;
    for (int x = 1;x < n;++x) {
        LL temp = 0;
        for (int d = 1;d <= (n - 1) / x;++d)
            temp = (temp + ((n - x * d) / __gcd(n - x * d, d) * d)) % MOD;
        ans = (ans + temp * f[x] % MOD) % MOD;
    }
    cout << ans << endl;

    return 0;
}