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 (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#define ll long long
#define rep(i,l,r)for(ll i=(l);i<(r);i++)
#define repp(i,l,r,k)for(ll i=(l);i<(r);i+=(k))
#define MOD 998244353

ll inv[1000010];
ll coe[1000010];
ll a[1000010];
int main(){
	inv[1]=1;
	rep(i,2,1000001)inv[i]=inv[MOD%i]*(MOD-MOD/i)%MOD;
	rep(i,1,1000001){
		coe[i]=(coe[i]+inv[i])%MOD;
		repp(j,2*i,1000001,i)coe[j]-=coe[i];
	}

	int n;
	ll ans=0;
	scanf("%d",&n);
	rep(i,0,n){
		int t;
		scanf("%d",&t);
		a[t]++;
		ans-=t;
	}
	rep(d,1,1000001){
		ll sum=0;
		repp(i,d,1000001,d)sum+=i*a[i];
		sum%=MOD;
		ans+=sum*sum%MOD*coe[d]%MOD;
	}
	ans%=MOD;
	if(ans<0)ans+=MOD;
	printf("%lld",ans*inv[2]%MOD);
}