「luogu - P3911」最小公倍数之和

link。

Denote \(cnt_{x}\) = the number of occurrences of \(x\), \(h\) = the maximum of \(a_i\), there we get

\[\sum_{1\leqslant i\leqslant n}\sum_{1\leqslant j\leqslant n}[a_i,a_j] \\
\begin{aligned}
&=\sum_{1\leqslant d\leqslant h}\sum_{1\leqslant i\leqslant h}\sum_{1\leqslant j\leqslant h}[\left(i,j\right)=d]\times\frac{i\times j\times cnt_i\times cnt_j}{d} \\
&=\sum_{1\leqslant d\leqslant h}d\sum_{1\leqslant i\leqslant\lfloor\frac{h}{d}\rfloor}\sum_{1\leqslant j\leqslant\lfloor\frac{h}{d}\rfloor}\sum_{k\mid\left(i,j\right)}\mu\left(k\right)\times i\times j\times cnt_{id}\times cnt_{jd} \\
&=\sum_{1\leqslant d\leqslant h}d\sum_{1\leqslant k\leqslant\lfloor\frac{h}{d}\rfloor}\mu\left(k\right)\times k^2\sum_{1\leqslant i\leqslant\lfloor\frac{h}{dk}\rfloor}\sum_{1\leqslant j\leqslant\lfloor\frac{h}{dk}\rfloor}i\times j\times cnt_{idk}\times cnt_{jdk} \\
&=\sum_{1\leqslant T\leqslant h}T\sum_{k\mid T}\mu\left(k\right)\times k\sum_{1\leqslant i\leqslant \lfloor\frac{h}{T}\rfloor}i\times cnt_{iT}\sum_{1\leqslant j\leqslant \lfloor\frac{h}{T}\rfloor}j\times cnt_{jT} \\
\end{aligned}
\]

Denote \(\displaystyle f(T)=\sum_{1\leqslant i\leqslant h/T}i\times cnt_{iT}\), \(\displaystyle g(T)=\sum_{1\leqslant i\leqslant h/T}i\times cnt_{iT}\times f(T)\)

\[\sum_{1\leqslant T\leqslant h}T\sum_{k\mid T}\mu\left(k\right)\times k\sum_{1\leqslant i\leqslant \lfloor\frac{h}{T}\rfloor}i\times cnt_{iT}\sum_{1\leqslant j\leqslant \lfloor\frac{h}{T}\rfloor}j\times cnt_{jT} \\
\begin{aligned}
&=\sum_{1\leqslant T\leqslant h}T\sum_{k\mid T}\mu\left(k\right)\times k\times g\left(T\right)
\end{aligned}
\]

Denote \(\displaystyle z(T)=T\sum_{k\mid T}\mu(k)\times k\), the answer would be \(\displaystyle\sum_{1\leqslant i\leqslant h}z(i)\times g(i)\).

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define cmin(x, y) x = min(x, y)
#define cmax(x, y) x = max(x, y)
#define fors(i, l, r, ...) for(int i = (l), REP##i = (r), ##__VA_ARGS__; i <= REP##i; ++i)
#define dfors(i, r, l, ...) for(int i = (r), REP##i = (l), ##__VA_ARGS__; i >= REP##i; --i)
signed main() {
    ios::sync_with_stdio(0), cin.tie(0);
    int n;
    cin >> n;
    vector<int> a(n);
    for(int& x : a) cin >> x;
    const int h = *max_element(a.begin(), a.end());
    vector<int> cnt(h+1);
    for(const int x : a) cnt[x]++;
    const auto z = [](const int n) {
        vector<int> mu(n+1), prime, not_prime(n+1);
        vector<ll> z(n+1);
        mu[1] = 1;
        fors(i, 2, n) {
            if(not_prime[i] == 0) prime.emplace_back(i),mu[i] = -1;
            for(const int p : prime) {
                if(i > n/p) break;
                not_prime[i*p] = 1;
                if(i%p == 0) break;
                mu[i*p] = -mu[i];
            }
        }
        fors(d, 1, n) {
            for(int T = d; T <= n; T += d) z[T] += ll(mu[d])*d;
        }
        return z;
    }(h);
    ll ans = 0;
    fors(i, 1, h) {
        ll tmp = 0;
        fors(j, 1, h/i) tmp += ll(cnt[i*j])*j;
        ans += tmp*tmp*z[i]*i;
    }
    cout << ans << "\n";
    return 0;
}