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
def modpow(a, n, mod)
  exp = Math.log(n, 2).floor
  rep = modpow_cl(a, n, exp, mod)
  return rep[0]
end

def modpow_cl(a, n, exp, mod)
  if exp < 0
    rep = [1, a]
  else
    if n >= 2**exp
      rep = modpow_cl(a, n-2**exp, exp-1, mod)
      rep[0] = rep[0]*rep[1] % mod
    else
      rep = modpow_cl(a, n, exp-1, mod)
    end
    rep[1] = rep[1]**2 % mod
  end
  return rep
end

n, k = gets.split.map(&:to_i)
cnt = Array.new(k+1, 0)

num = k
while num > 0
  cnt[num] = modpow(k/num, n, MOD)
  id = num*2
  while id <= k
    cnt[num] = (cnt[num]-cnt[id]) % MOD
    id += num
  end
  num -= 1
end

sum = 0
for i in 1..k
  sum = (sum + cnt[i]*i) % MOD
end
puts sum