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 MX = 100005;
  const long long mod = 1e9 + 7;
  int phi[MX];
  void precalc_phi() {
    phi[0] = 0;
    phi[1] = 1;
    for (int i = 2; i < MX; i++) phi[i] = i;
    for (int i = 2; i < MX; i++)
      if (phi[i] == i)
        for (int j = i; j < MX; j += i) phi[j] -= phi[j] / i;
  }
  vector<int> d[MX];
  void precalc_divisors() {
    for (int i = 1; i < MX; i++)
      for (int j = i; j < MX; j += i) d[j].push_back(i);
  }
  int n, arr[MX], cnt[MX];
  int main() {
    cin.tie(0)->sync_with_stdio(0);
    precalc_phi();
    precalc_divisors();
    cin >> n;
    for (int i = 1; i <= n; i++) cin >> arr[i];
    long long ans = 0ll;
    for (int x = 1; x <= n; x++) {
      for (int i = x; i <= n; i += x)
        for (int j : d[arr[i]]) cnt[j]++;
      for (int i = x; i <= n; i += x)
        for (int j : d[arr[i]])
          if (cnt[j]) {
            (ans += phi[x] *
                    (phi[j] * 1ll * (cnt[j] * 1ll * cnt[j] % mod) % mod) % mod) %=
                mod;
            cnt[j] = 0;
          }
    }
    cout << ans << endl;
    return 0;
  }
  