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)

{-# LANGUAGE BangPatterns #-}
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as UM
import qualified Data.ByteString.Char8 as C
import Data.List

r' :: IO [Int]
r' = unfoldr (C.readInt . C.dropWhile (==' ')) <$> C.getLine

-- editorial
main = do
    [n,k] <- r'
    xs <- UM.replicate (k+1) 0
    calc <- U.freeze =<< tab n k k xs
    print $ U.foldr1' (+%) $ U.imap (*%) calc

over n k x = (div k x) ^% n

tab n k 0 xs = return xs
tab n k x xs = do
    y <- foldr (+%) 0 <$> mapM (UM.read xs) [2*x,3*x..k]
    UM.write xs x (over n k x -% y)
    tab n k (x-1) xs


mdl = 1000000007

(^%) :: Int -> Int -> Int
b ^% e = powAux (b `rem` mdl) e 1 `mod` mdl
    where 
        powAux _ 0 !ac = ac
        powAux b e !ac = powAux (b*b `rem` mdl) (e `quot` 2)
            (if odd e then b*ac `rem` mdl else ac)

(+%), (-%), (*%), (/%) :: Int -> Int -> Int
a +% b = (a + b) `rem` mdl
a -% b = (a - b) `mod` mdl
a *% b = (a * b) `rem` mdl
a /% b = (a * invMod b) `mod` mdl
    where   invMod :: Int -> Int
            invMod = (^% (mdl-2))