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;
typedef long long ll;
const int P = 998244353;
const int N = 1000000 + 1;
int n, x, c[N], c2[N];
ll f[N];
int inv[N];
void get_inv() {
    inv[1] = 1;
    for (int i = 2; i < N; i++) inv[i] = 1ll * inv[P % i] * (P - P / i) % P;
}
int main() {
    get_inv();
    scanf("%d", &n);
    for (int i = 1; i <= n; i++) {
        scanf("%d", &x);
        c[x] = (c[x] + x) % P;
        c2[x] = (c2[x] + 1ll * x * x % P) % P;
    }
    for (int i = 1; i < N; i++) {
        for (int j = i + i; j < N; j += i) {
            c[i] = (c[i] + c[j]) % P;
            c2[i] = (c2[i] + c2[j]) % P;
        }
        //printf("%d %d\n",c[i],c2[i]);
        f[i] =
            (1ll * c[i] * c[i] % P - c2[i] + P) % P * inv[2] % P;
    }
    for (int i = N - 1; i >= 1; --i)
        for (int j = i + i; j < N; j += i) f[i] = (f[i] - f[j] + P) % P;
    int ans = 0; 
    for (int i = 1; i < N; i++) ans = (ans + f[i] * inv[i] % P) % P;
    printf("%d\n", ans);
    return 0;
}