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 Python (3.8.2) (judge verdict: Accepted)

import sys

input = sys.stdin.readline
P = 10 ** 9 + 7


def main():
    N, K = map(int, input().split())

    n_multiple = [0] * (K + 1)
    for k in range(1, K + 1):
        n_multiple[k] = K // k

    n_gcd = [0] * (K + 1)
    for k in reversed(range(1, K + 1)):
        n_gcd[k] = pow(n_multiple[k], N, mod=P)
        m = 2
        kk = k * m
        while kk <= K:
            n_gcd[k] = (n_gcd[k] - n_gcd[kk]) % P
            m += 1
            kk = k * m

    ans = 0
    for k in range(1, K + 1):
        ans = (ans + k * n_gcd[k]) % P
    print(ans)


if __name__ == "__main__":
    main()