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)

# -*- coding: utf-8 -*-

from math import gcd

def main():
     n, k = map(int, input().split())
     N = pow(10, 9) + 7
     gcd_num = [0 for i in range(k+1)]
     ans = 0

     for i in range(k, 0, -1):
          x = k // i
          gcd_num[i] = pow(x, n, N)
          for j in range(2, x+1):
               gcd_num[i] -= gcd_num[j * i]
          
          ans += gcd_num[i] * i

     print(ans%N)

if __name__ == "__main__":
    main()