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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.IOException;
import java.io.UncheckedIOException;
import java.io.Closeable;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.InputStream;

/**
 * Built using CHelper plug-in Actual solution is at the top
 * 
 * @author daltao
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "daltao", 1 << 27);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            FastOutput out = new FastOutput(outputStream);
            LCMs solver = new LCMs();
            solver.solve(1, in, out);
            out.close();
        }
    }
    static class LCMs {
        NumberTheory.Modular mod = new NumberTheory.Modular(998244353);
        NumberTheory.Power pow = new NumberTheory.Power(mod);
        NumberTheory.Composite comp = new NumberTheory.Composite(1000000, mod);

        public void solve(int testNumber, FastInput in, FastOutput out) {
            int limit = 1000000;

            int n = in.readInt();
            int[] cnts = new int[limit + 1];
            int[] sum = new int[limit + 1];
            int[] pairSum = new int[limit + 1];
            for (int i = 0; i < n; i++) {
                cnts[in.readInt()]++;
            }
            for (int i = 1; i <= limit; i++) {
                pairSum[i] = mod.mul(comp.composite(cnts[i], 2), mod.mul(i, i));
                sum[i] = mod.mul(i, cnts[i]);
                for (int j = i + i; j <= limit; j += i) {
                    pairSum[i] = mod.plus(pairSum[i], mod.mul(comp.composite(cnts[j], 2), mod.mul(j, j)));
                    pairSum[i] = mod.plus(pairSum[i], mod.mul(sum[i], mod.mul(cnts[j], j)));
                    sum[i] = mod.plus(sum[i], mod.mul(cnts[j], j));
                }
            }


            int ans = 0;
            for (int i = limit; i >= 1; i--) {
                for (int j = i + i; j <= limit; j += i) {
                    pairSum[i] = mod.subtract(pairSum[i], pairSum[j]);
                }
                ans = mod.plus(ans, mod.mul(pairSum[i], pow.inverse(i)));
            }

            out.println(ans);
        }

    }
    static class FastOutput implements AutoCloseable, Closeable {
        private StringBuilder cache = new StringBuilder(10 << 20);
        private final Writer os;

        public FastOutput(Writer os) {
            this.os = os;
        }

        public FastOutput(OutputStream os) {
            this(new OutputStreamWriter(os));
        }

        public FastOutput println(int c) {
            cache.append(c).append('\n');
            return this;
        }

        public FastOutput flush() {
            try {
                os.append(cache);
                os.flush();
                cache.setLength(0);
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
            return this;
        }

        public void close() {
            flush();
            try {
                os.close();
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
        }

        public String toString() {
            return cache.toString();
        }

    }
    static class NumberTheory {
        public static class Modular {
            int m;

            public Modular(int m) {
                this.m = m;
            }

            public Modular(long m) {
                this.m = (int) m;
                if (this.m != m) {
                    throw new IllegalArgumentException();
                }
            }

            public Modular(double m) {
                this.m = (int) m;
                if (this.m != m) {
                    throw new IllegalArgumentException();
                }
            }

            public int valueOf(int x) {
                x %= m;
                if (x < 0) {
                    x += m;
                }
                return x;
            }

            public int valueOf(long x) {
                x %= m;
                if (x < 0) {
                    x += m;
                }
                return (int) x;
            }

            public int mul(int x, int y) {
                return valueOf((long) x * y);
            }

            public int plus(int x, int y) {
                return valueOf(x + y);
            }

            public int subtract(int x, int y) {
                return valueOf(x - y);
            }

            public String toString() {
                return "mod " + m;
            }

        }

        public static class Power {
            final NumberTheory.Modular modular;

            public Power(NumberTheory.Modular modular) {
                this.modular = modular;
            }

            public int pow(int x, long n) {
                if (n == 0) {
                    return 1;
                }
                long r = pow(x, n >> 1);
                r = modular.valueOf(r * r);
                if ((n & 1) == 1) {
                    r = modular.valueOf(r * x);
                }
                return (int) r;
            }

            public int inverse(int x) {
                return pow(x, modular.m - 2);
            }

        }

        public static class InverseNumber {
            int[] inv;

            public InverseNumber(int[] inv, int limit, NumberTheory.Modular modular) {
                this.inv = inv;
                inv[1] = 1;
                int p = modular.m;
                for (int i = 2; i <= limit; i++) {
                    int k = p / i;
                    int r = p % i;
                    inv[i] = modular.mul(-k, inv[r]);
                }
            }

            public InverseNumber(int limit, NumberTheory.Modular modular) {
                this(new int[limit + 1], limit, modular);
            }

        }

        public static class Factorial {
            int[] fact;
            int[] inv;
            NumberTheory.Modular modular;

            public Factorial(int[] fact, int[] inv, NumberTheory.InverseNumber in, int limit,
                            NumberTheory.Modular modular) {
                this.modular = modular;
                this.fact = fact;
                this.inv = inv;
                fact[0] = inv[0] = 1;
                for (int i = 1; i <= limit; i++) {
                    fact[i] = modular.mul(fact[i - 1], i);
                    inv[i] = modular.mul(inv[i - 1], in.inv[i]);
                }
            }

            public Factorial(int limit, NumberTheory.Modular modular) {
                this(new int[limit + 1], new int[limit + 1], new NumberTheory.InverseNumber(limit, modular), limit,
                                modular);
            }

        }

        public static class Composite {
            final NumberTheory.Factorial factorial;
            final NumberTheory.Modular modular;

            public Composite(NumberTheory.Factorial factorial) {
                this.factorial = factorial;
                this.modular = factorial.modular;
            }

            public Composite(int limit, NumberTheory.Modular modular) {
                this(new NumberTheory.Factorial(limit, modular));
            }

            public int composite(int m, int n) {
                if (n > m) {
                    return 0;
                }
                return modular.mul(modular.mul(factorial.fact[m], factorial.inv[n]), factorial.inv[m - n]);
            }

        }

    }
    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    bufLen = -1;
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }
}