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)

n,k=gets.split(' ').map(&:to_i)
#h=gets.split(' ').map(&:to_i)
#tajigen = Array.new(4) { Array.new(2,0) }
#maze = h.times.map { gets.chomp }
wari = Array.new(k+1)
kkk=10**9+7
=begin
for i in 1..k
  #puts i
  wari[i] = (k/i)**n
end
#puts wari
=end
ans=0
=begin
def power2(a, n)
  p = 1
  n.to_s(2).split('').each{|i|
    p *= p

    if i == '1'
      p *= a
    end
  }
  p
end
=end

def pow(x, n, mod)
  return 1 if n == 0
  return pow(x**2 % mod, n / 2, mod) if (n % 2) == 0
  return pow(x**2 % mod, n / 2, mod) * x % mod if (n % 2) == 1
end
#puts power2(a,n)
k.downto(1) do |soto|
  wari[soto] = pow(k/soto,n,kkk)  #((k/soto)**n)
  count=2
  while soto * count <= k
    wari[soto]-=wari[soto * count]
    count+=1
  end
  ans += (wari[soto]*soto)
  #puts wari[soto]
end
puts ans%kkk
=begin
puts "iii"
wari[0] = 0
for i in 0..(k/2-1)
  put = k/2-i
  count=2
  #puts i
  while put * count <= k
    wari[put]=wari[put] - wari[put * count]
    count+=1
  end
end
=end
#puts wari
=begin
ans=0
for i in 1..k
  ans += (i * wari[i]) % kkk
  ans %= kkk
#  ans = ans%kkk
end
=end
#puts ans