Problem:
Score : 700 points

Problem Statement

We have an integer sequence of length N: A_0,A_1,\cdots,A_{N-1}.

Find the following sum (\mathrm{lcm}(a, b) denotes the least common multiple of a and b):

\sum_{i=0}^{N-2} \sum_{j=i+1}^{N-1} \mathrm{lcm}(A_i,A_j)

Since the answer may be enormous, compute it modulo 998244353.

Constraints

1 \leq N \leq 200000

1 \leq A_i \leq 1000000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0\ A_1\ \cdots\ A_{N-1}

Output

Print the sum modulo 998244353.

Sample Input 1

3
2 4 6

Sample Output 1

22

\mathrm{lcm}(2,4)+\mathrm{lcm}(2,6)+\mathrm{lcm}(4,6)=4+6+12=22.

Sample Input 2

8
1 2 3 4 6 8 12 12

Sample Output 2

313

Sample Input 3

10
356822 296174 484500 710640 518322 888250 259161 609120 592348 713644

Sample Output 3

353891724

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

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

const int MX = 1E6 + 5, MOD = 998244353;

int n, u;
long long ans = 0, f[MX], inv[MX], sum[MX];

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    inv[1] = 1;
    for (int i = 2; i < MX; i++) {
        inv[i] = MOD - MOD / i * inv[MOD % i] % MOD;
    }
    cin >> n;
    while (n--) {
        cin >> u;
        (ans -= u) %= MOD;
        (sum[u] += u) %= MOD;
    }
    for (int i = MX - 1; i >= 1; i--) {
        long long tot = 0;
        for (int j = i; j < MX; j += i) {
            (tot += sum[j]) %= MOD;
            (f[i] -= f[j]) %= MOD;
        }
        (f[i] += tot * tot % MOD) %= MOD;
        (ans += f[i] * inv[i]) %= MOD;
    }
    (ans *= inv[2]) %= MOD;
    cout << (ans + MOD) % MOD;
}