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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import Data.Array

solve :: Int -> Int -> Mint
solve n k = sum $ map (\(i,a)->M i*a) $ assocs arr where
  arr = array (1, k) [ (i, M (div k i) ^ n - sum ext) |
    i <- [k, k-1 .. 1],
    let ext = [arr!(i*m) | m <- [2..div k i] ] ]

main = do
  [n, k] <- map read.words <$> getLine
  print $ solve n k

modulus = 10^9 + 7
newtype Mint = M Int
instance Num Mint where
  (M a) + (M b)
    | a + b < modulus = M $ a + b
    | otherwise = M $ a + b - modulus
  (M a) * (M b) = M $ mod (a * b) modulus
  negate (M 0) = M 0
  negate (M a) = M $ modulus - a
  fromInteger a = M $ mod (fromInteger a) modulus
  abs = id
  signum _ = M 1
instance Show Mint where
  show (M a) = show a