description: Mr. Chanek has an array a of n integers. The prettiness value of a is denoted as:
  
  $$$∑_{i=1}^{n} {∑_{j=1}^{n} {\gcd(a_i, a_j) ⋅ \gcd(i, j)}}$$$
  
  where \gcd(x, y) denotes the greatest common divisor (GCD) of integers x and y.
  
  In other words, the prettiness value of an array a is the total sum of \gcd(a_i, a_j) ⋅ \gcd(i, j) for all pairs (i, j).
  
  Help Mr. Chanek find the prettiness value of a, and output the result modulo 10^9 + 7!
  
  Input
  
  The first line contains an integer n (2 ≤ n ≤ 10^5).
  
  The second line contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 10^5).
  
  Output
  
  Output an integer denoting the prettiness value of a modulo 10^9 + 7.
  
  Example
  
  Input
  
  
  5
  3 6 2 1 4
  
  
  Output
  
  
  77
solution: #include <bits/stdc++.h>
  using namespace std;
  const int N = 1e5 + 5, mod = 1e9 + 7;
  int n, a[N], phi[N], cnt[N];
  vector<int> d[N];
  void init() {
    for (int i = 1; i < (N); ++i) {
      phi[i] = i;
    }
    for (int i = 2; i < (N); ++i) {
      if (phi[i] == i) {
        for (int j = i; j < N; j += i) {
          phi[j] -= phi[j] / i;
        }
      }
    }
    for (int i = 1; i < (N); ++i) {
      for (int j = i; j < N; j += i) {
        d[j].push_back(i);
      }
    }
  }
  int main() {
    cin.tie(0)->sync_with_stdio(0);
    cin.exceptions(cin.failbit);
    init();
    cin >> n;
    for (int i = 1; i < (n + 1); ++i) {
      cin >> a[i];
    }
    long long ans = 0;
    for (int x = 1; x < (n + 1); ++x) {
      for (int i = x; i <= n; i += x) {
        for (const int j : d[a[i]]) {
          ++cnt[j];
        }
      }
      for (int i = x; i <= n; i += x) {
        for (const int j : d[a[i]]) {
          if (cnt[j]) {
            ans = (ans +
                   phi[x] * 1ll * phi[j] % mod * cnt[j] % mod * cnt[j] % mod) %
                  mod;
            cnt[j] = 0;
          }
        }
      }
    }
    cout << ans << '\n';
  }
  