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)

mod = 10 ** 9 + 7

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

a = [1] * (k+1)
b = Array.new(k+1){|i| i }

ans = 0
(k/2).downto(1) do |i|
  cnt = ( k/i ).pow( n, mod )
  (i*2).step(k, i){|j| cnt = ( cnt - a[j] ) % mod }
  a[i] = cnt
  ans += i * cnt % mod
  ans %= mod if ans >= mod
end

puts (ans + (k-k/2) * ( k + k/2+1) / 2 ) % mod