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 MAX = 1e6, MOD = 998244353;
int add(int a, int b) { return (a + b) % MOD; }
int mul(int a, int b) { return 1LL * a * b % MOD; }

int n, a[200000], sum[MAX + 1], w[MAX + 1], inv[MAX + 1];
int main() {
    scanf("%d", &n);
    int ans = 0;
    for (int i = 0; i < n; ++i) {
        scanf("%d", a + i);
        sum[a[i]] = add(sum[a[i]], a[i]);
        ans = add(ans, -a[i]);
    }

    inv[1] = 1;
    for (int i = 2; i <= MAX; ++i) inv[i] = -mul(MOD / i, inv[MOD % i]);

    for (int d = 1; d <= MAX; ++d) {
        w[d] = add(w[d], inv[d]);
        int s = sum[d];
        for (int e = 2 * d; e <= MAX; e += d) s = add(s, sum[e]), w[e] = add(w[e], -w[d]);
        ans = add(ans, mul(w[d], mul(s, s)));
    }

    ans = mul(ans, inv[2]);
    if (ans < 0) ans += MOD;
    printf("%d\n", ans);
}