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>
#define MOD 998244353
#define inv2 499122177

using namespace std;

typedef long long ll;

const int Maxn=1000000;

int prime[200000],mu[Maxn+5];
bool check[Maxn+5];

ll sum[Maxn+5];

void pre() {
  mu[1]=1;
  int tot=0;
  for(int i=2;i<=Maxn;i++) {
  	if (!check[i]) {
  		prime[++tot]=i;
  		mu[i]=MOD-1;
	  }
	for(int j=1;j<=tot&&i*prime[j]<=Maxn;j++) {
		check[i*prime[j]]=1;
		if (i%prime[j]==0) break;
		mu[i*prime[j]]=(MOD-mu[i])%MOD;
	}
  }
  for(int i=1;i<=Maxn;i++)
    for(int j=1;i*j<=Maxn;j++) sum[i*j]=(sum[i*j]+(ll)i*j%MOD*j%MOD*mu[j])%MOD;
}

int cnt[Maxn+5];

int main() {
  pre();
  int n;
  scanf("%d",&n);
  int s=0;
  for(int i=1;i<=n;i++) {
  	int x;
  	scanf("%d",&x);
  	cnt[x]++;
  	s=(s+x)%MOD;
  }
  ll ans=0;
  for(int i=1;i<=Maxn;i++) {
  	ll s=0;
  	for(int j=1;i*j<=Maxn;j++) s=(s+(ll)cnt[i*j]*j)%MOD;
  	ans=(ans+sum[i]*s%MOD*s)%MOD;
  }
  printf("%lld\n",(ans-s+MOD)*inv2%MOD);
  return 0;
}