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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include<iostream>
#include<vector>

using i64 = long long;

constexpr i64 mod = 1000000007;

i64 pow(i64 a, i64 b) {
  i64 r = 1;
  while (b != 0) {
    if (b % 2 != 0) {
      r = r * a % mod;
    }
    a = a * a % mod;
    b /= 2;
  }
  return r;
}

int main() {
  i64 n, k;
  std::cin >> n >> k;
  std::vector<i64> a(k + 1);
  for (int i = 1; i <= k; i += 1) {
    a[i] = pow(k / i, n);
  }
  
  for (int i = k; i >= 1; i -= 1) {
    for (int j = i * 2; j <= k; j += i) {
      a[i] = (a[i] - a[j] + mod) % mod;
    }
  }
  
  i64 ans = 0;
  for (int i = 1; i <= k; i += 1) {
    ans = (ans + a[i] * i % mod) % mod;
  }
  
  std::cout << ans << "\n";
}