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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

import sys
import numpy as np


def prepare(n, MOD):
    f = 1
    factorials = [1]
    for m in range(1, n + 1):
        f *= m
        f %= MOD
        factorials.append(f)
    inv = pow(f, MOD - 2, MOD)
    invs = [1] * (n + 1)
    invs[n] = inv
    for m in range(n, 1, -1):
        inv *= m
        inv %= MOD
        invs[m - 1] = inv

    solo_invs = [0] + [f * i % MOD for f, i in zip(factorials, invs[1:])]

    return factorials, invs, solo_invs


def decompose_inverses(solo_invs, MOD):
    # 各整数 g に対して、g の約数である各 i について dcm[i] を全て足すと 1/g になるような数列を作成
    dcm = np.array(solo_invs, dtype=np.int64)
    for i in range(1, len(dcm)):
        dcm[2 * i::i] -= dcm[i]
    dcm %= MOD
    return dcm


n, *aaa = map(int, sys.stdin.buffer.read().split())
MOD = 998244353
cnt = np.bincount(aaa)
LIMIT = len(cnt) - 1
facts, invs, solo_invs = prepare(LIMIT, MOD)
dcm = decompose_inverses(solo_invs, MOD)
idx = np.arange(LIMIT + 1, dtype=np.int64)
base1 = idx * cnt
base2 = idx ** 2 * cnt

ans = 0
inv2 = solo_invs[2]
for d in range(1, LIMIT + 1):
    ans = (ans + int(dcm[d]) * int(base1[d::d].sum() ** 2 - base2[d::d].sum()) * inv2) % MOD
print(ans)