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 = 1000000
X = [-1] * (N+1)
L = [{} for _ in range(N+1)]
k = 2
while k <= N:
    X[k] = 1
    L[k][k] = 1
    for i in range(k*2, N+1, k):
        X[i] = 0
        L[i][k] = 1
    d = 2
    while k**d <= N:
        for i in range(k**d, N+1, k**d):
            L[i][k] += 1
        d += 1
    while k <= N and X[k] >= 0:
        k += 1
P = [i for i in range(N+1) if X[i] == 1]