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 $998244353$.

3
2 4 6

22

$\mathrm{lcm}(2,4)+\mathrm{lcm}(2,6)+\mathrm{lcm}(4,6)=4+6+12=22$.

8
1 2 3 4 6 8 12 12

313

10
356822 296174 484500 710640 518322 888250 259161 609120 592348 713644

353891724