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 PyPy3 (7.3.0) (judge verdict: Wrong Answer)

import math
from functools import reduce
from sys import stdin
import itertools

def gcd(*numbers):
    return reduce(math.gcd, numbers)
  
def main():
  N, K = map(int, stdin.readline().split(" "))
  
  ans = 0
  
  for xs in itertools.combinations_with_replacement(range(1, K+1), N):
    
    if len(set(xs)) == 1:
      ans += gcd(*xs)
    elif len(set(xs)) == 2:
      ans += 3*gcd(*xs)
    else:
      ans += 6*gcd(*xs)
      
  print(ans % (10**9 + 7))

if __name__ == '__main__':
  main()