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++14 (GCC 5.4.1) (judge verdict: Accepted)

// #pragma GCC target("sse,sse2,sse3,ssse3,sse4")
#pragma GCC optimize("Ofast")
// #pragma GCC optimize("unroll-loops")
#include <list>
 
// hloya template v25
  
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// ░░░░░░█░█░░░░░░░░▄▀▀▄░▀░█░█▄▀▀▄░
// █▀▀█▄░█░█░░▀░░░░░█░░░▀▄▄█▄▀░░░█░
// ▀▄▄░▀██░█▄░▀░░░▄▄▀░░░░░░░░░░░░▀▄
// ░░▀█▄▄█░█░░░░▄░░█░░░▄█░░░▄░▄█░░█
// ░░░░░▀█░▀▄▀░░░░░█░██░▄░░▄░░▄░███
// ░░░░░▄█▄░░▀▀▀▀▀▀▀▀▄░░▀▀▀▀▀▀▀░▄▀░
// ░░░░█░░▄█▀█▀▀█▀▀▀▀▀▀█▀▀█▀█▀▀█░░░
// ░░░░▀▀▀▀░░▀▀▀░░░░░░░░▀▀▀░░▀▀░░░░
  
#include <bits/stdc++.h>
#include <valarray>
using namespace std;
  
bool dbg = 0;
  
clock_t start_time = clock();
#define current_time fixed<<setprecision(6)<<(ld)(clock()-start_time)/CLOCKS_PER_SEC
  
#define f first
#define s second
#define mp make_pair
#define mt make_tuple
#define pb push_back
#define eb emplace_back
#define all(x) (x).begin(), (x).end()
  
#define ll long long
#define ld long double
#define pii pair<int,int>
#define umap unordered_map<int, int>
  
#define files1 freopen("input.txt","r",stdin)
#define files2 freopen("out2.txt","w",stdout)
#define files files1;files2
#define fast_io ios_base::sync_with_stdio(0);cin.tie(0)
  
 
// #define endl '\n'
#define ln(i,n) " \n"[(i) == (n) - 1]
  
void bad(string mes = "Impossible"){cout << mes;exit(0);}
void bad(int mes){cout << mes;exit(0);}
  
template<typename T>
string bin(T x, int st = 2){
    string ans = "";
    while (x > 0){
        ans += char('0' + x % st);
        x /= st;
    }
    reverse(ans.begin(), ans.end());
    return ans.empty() ? "0" : ans;
}
  
template<typename T>
void upmax(T& x, T y) {
    x = max(x, y);
}
  
template<typename T>
bool upmin(T& x, T y) {
    if (x > y) {
        x = y;
        return true;
    }
    return false;
}
 
// inline int popcount(int x){
//     int count = 0;
//     __asm__ volatile("POPCNT %1, %0;":"=r"(count):"r"(x):);
//     return count;
// }
  
template<typename T>
T input(){
    T ans = 0, m = 1;
    char c = ' ';
  
    while (!((c >= '0' && c <= '9') || c == '-')) {
        c = getchar();
    }
  
    if (c == '-')
        m = -1, c = getchar();
    while (c >= '0' && c <= '9'){
        ans = ans * 10 + (c - '0'), c = getchar();
    }
    return ans * m;
}
  
template<typename T> void read(T& a) { a = input<T>(); }
template<typename T> void read(T& a, T& b) { read(a), read(b); }
template<typename T> void read(T& a, T& b, T& c) { read(a, b), read(c); }
template<typename T> void read(T& a, T& b, T& c, T& d) { read(a, b), read(c, d); }
  
const int inf = 2e9 + 20;
const short short_inf = 3e4 + 20;
const long double eps = 1e-6;
const int maxn = (int)1e6 + 12, base = 1e9 + 7;
const ll llinf = 2e18 + 5;
const double PI = acos(-1.0);
 
const int mod = 998244353;
  
template<typename T>
T binpow(T n, T s)
{
    if (s <= 0)
        return 1LL;
    if (s % 2 == 0){
        T b = binpow(n, s / 2);
        return ( 1LL * b * b ) % mod;
    } else {
        return (1LL* binpow(n, s - 1) * n) % mod;
    }
}

int a[maxn], cnt[maxn], res[maxn];

int main() {
    fast_io;
    // files1;

    int n;
    cin >> n;

    int ans = 0;
    for (int i = 0; i < n; i++) {
        cin >> a[i];
        ans += 1ll * cnt[a[i]] * a[i] % mod;
        ans %= mod;
        cnt[a[i]]++;
    }

    for (int g = 1; g < maxn; g++) {
        int sum = 0;
        for (int i = g; i < maxn; i += g) {
            res[g] += 1ll * sum * cnt[i] % mod * i % mod;
            res[g] %= mod;
            sum += 1ll * cnt[i] * i % mod;
            sum %= mod;
        }
        // if (res[g]) {   
        //     cout << g << ' ' << res[g] << endl;
        // }
    }

    for (int g = maxn - 1; g; g--) {
        for (int i = g + g; i < maxn; i += g) {
            res[g] -= res[i];
            if (res[g] < 0)
                res[g] += mod;
        }
        ans += 1ll * res[g] * binpow(g, mod - 2) % mod;
        ans %= mod;
    }
    cout << ans;
    return 0;
}