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 Ruby (2.7.1) (judge verdict: Accepted)

def modpow(a, n, m)
  return 1 if n == 0

  tmp = modpow(a, n / 2, m)
  n % 2 == 0 ? tmp * tmp % m : tmp * tmp * a % m
end

M = 1_000_000_007

n, k = gets.chomp.split(' ').map(&:to_i)

g = (0..k).map { 0 }

answer = k.downto(1).map { |m|
  g[m] = (k / m).downto(2).inject(modpow(k / m, n, M)) { |r, i| (r - g[i * m]) % M } 
  g[m] * m % M
}.sum % M

puts answer