Greek Casino

#include <bits/stdc++.h>
#ifdef ENABLE_DEBUG
  #define DEBUG(x) std::cout << x << std::endl
#else
  #define DEBUG(x)
#endif
#define fi first
#define se second
#define pb push_back
#define all(x) x.begin(),x.end()
#define rall(x) x.rbegin(),x.rend()
using namespace std;
using ii = pair<int, int>;
using i64 = long long;
using ld = long double;
const int INF = 0x3f3f3f3f;
const i64 INFLL = 0x3f3f3f3f3f3f3f3f;
const int N = 1e5;

vector<vector<int>> dv;
vector<vector<int>> fac;

int main() {
  cin.tie(0)->sync_with_stdio(0);

  dv.assign(N+1, vector<int>());
  fac.assign(N+1, vector<int>());
  vector<int> cnt(N+1, 0);

  for (int i = 1; i <= N; ++i) {
    int l = i;
    for (int j = 2; (i64)j * j <= l; ++j) {
      while (l % j == 0) {
        l /= j;
        fac[i].pb(j);
      }
    }
    if (l > 1) fac[i].pb(l);
    for (int j = i; j <= N; j += i) {
      dv[j].pb(i);
    }
  }
  int n; cin >> n;

  vector<ld> a(n + 1, 0.0);
  vector<int> b(n + 1);

  int sum = 0;
  for (int i = 1; i <= n; ++i) {
    cin >> b[i];
    sum += b[i];
  }

  for (int i = 1; i <= n; ++i) {
    a[i] = (ld) 1.0 * b[i] / (ld) sum;
  }

  vector<ld> dp(n + 1, 0.0);
  for (int x = n; x >= 1; --x) {
    vector<ii> caras;

    for (int y = x; y <= n; y += x) {
      // x | y

      int f = 1;

      for (int d : fac[x]) {
        ++cnt[d];
      }

      int xx = x;

      int last = -1;
      int counter = 0;
      int prod = 1;

      for (int d : fac[y]) {
        if (d == last) {
          prod *= d;
          ++counter;
        } else {
          if (last != -1 and counter > cnt[last]) {
            f *= prod;
            for (int i = 0; i < cnt[last]; ++i)
              xx /= last;
          }
          prod = d;
          counter = 1;
        }
        last = d;
      }

      if (last != -1 and counter > cnt[last]) {
        f *= prod;
        for (int i = 0; i < cnt[last]; ++i)
          xx /= last;
      }

      for (int d : dv[xx]) if (d * f <= n) {
        int z = d * f;
        caras.pb({y, z});
      } else {
        break;
      }

      for (int d : fac[x]) {
        --cnt[d];
      }
    }

    ld s1 = 0;
    ld ss = 0;
    for (auto [z, j] : caras) {
      if (z == x) {
        ss += a[j];
        s1 += a[j];
      } else {
        ss += (ld) a[j] * (1.0 + dp[z]);
      }
    }

    dp[x] = ss / ld(1.0 - s1);
  }

  cout << fixed << setprecision(10) << dp[1] << '\n';
}