Problem:
Score : 500 points

Problem Statement

Consider sequences \{A_1,...,A_N\} of length N consisting of integers between 1 and K (inclusive).

There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.

Since this sum can be enormous, print the value modulo (10^9+7).

Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.

Constraints

2 \leq N \leq 10^5

1 \leq K \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).

Sample Input 1

3 2

Sample Output 1

9

\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9

Thus, the answer is 9.

Sample Input 2

3 200

Sample Output 2

10813692

Sample Input 3

100000 100000

Sample Output 3

742202979

Be sure to print the sum modulo (10^9+7).

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)


import java.util.Scanner;

public class Main {
    private static final long MOD = 1_000_000_007;

    public static void main(final String[] args) {
        final Scanner scanner = new Scanner(System.in);
        final int n = scanner.nextInt();
        final int k = scanner.nextInt();

        long sum = 0;
        final long[] array = new long[k + 1];
        for (int i = k; i >= 1; i--) {
            final int nums = k / i;
            long multi = iterativePow(nums, n);

            for (int j = 2; i * j <= k; j++) {
                multi = (multi + MOD - array[i * j]) % MOD;
            }

            sum += (multi * i) % MOD;
            sum %= MOD;
            array[i] = multi;
        }
        System.out.println(sum);
    }

    private static long iterativePow(long a, long b) {
        if (a == 1) {
            return 1;
        }
        long tmp = 1;
        while (b > 0) {
            if ((b & 1) == 1) {
                tmp *= a;
            }
            a *= a;
            a %= MOD;
            b >>= 1;
            tmp %= MOD;
        }

        return tmp;
    }
}