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.Maybe
import Data.Array
import Data.List
import qualified Data.ByteString.Char8 as C

parseInt :: C.ByteString -> Int
parseInt = fst . fromJust . C.readInt

m = 10^9 + 7
x @* y = x * y `mod` m
x @+ y = let z = x + y in if z >= m then z - m else z

fpm :: Int -> Int ->Int
fpm a e
  | e == 0 = 1
  | odd e = a' @* a' @* a
  | otherwise = a' @* a'
  where
  a' = fpm a (e `div` 2)

main :: IO ()
main = do
  [n, k] <- map parseInt . C.words <$> C.getLine
  let calc = foldl' (@+) 0
      f = array (1, k) [(i, fpm y n - calc [f ! (j * i) | j <- [2..y]]) | i <- reverse [1..k], let y = k `div` i]
  putStrLn $ show $ calc [f ! i @* i | i <- [1..k]]