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 PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

N=int(input())
A=list(map(int,input().split()))
mod=998244353

inverse = [0, 1]
for i in range( 2, 10**6 + 1 ):
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )

data={i:[0,0] for i in range(1,10**6+1)}
for i in range(N):
    data[A[i]][0]+=A[i]
    data[A[i]][1]+=A[i]**2

for i in range(1,10**6+1):
    for j in range(1,10**6//i+1):
        data[i][0]+=data[i*j][0]
        data[i][1]+=data[i*j][1]

ans={i:0 for i in data}
for i in range(10**6,0,-1):
    for j in range(1,10**6//i+1):
        ans[i]-=ans[i*j]
        ans[i]%=mod
    ans[i]+=(data[i][0]**2-data[i][1])//2
    ans[i]%=mod

res=0
for i in ans:
    res+=inverse[i]*ans[i]
    res%=mod

print(res)