Problem:
Score : 400 points

Problem Statement

You are given positive integers N and M.

How many sequences a of length N consisting of positive integers satisfy a_1 \times a_2 \times ... \times a_N = M? Find the count modulo 10^9+7.

Here, two sequences a' and a'' are considered different when there exists some i such that a_i' \neq a_i''.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

1 \leq M \leq 10^9

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo 10^9 + 7.

Sample Input 1

2 6

Sample Output 1

4

Four sequences satisfy the condition: \{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\} and \{6, 1\}.

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001

Solution in Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

import Control.Monad
import qualified Data.IntSet as IntSet
import Data.Maybe
import Debug.Trace
import Data.List
import qualified Data.IntMap.Strict as IntMap

main :: IO ()
main = do
  (n:m:_) <- fmap (map read . words) getLine :: IO [Int]
  print $ solve n m

solve :: Int -> Int -> Int
solve n m = foldr (\x acc -> acc * (countNum x n)) 1 counts
  where
    as = factorization m
    counts = IntSet.foldr (\x acc -> IntMap.insert x (length $ filter (==x) as) acc) IntMap.empty $ IntSet.fromList as

divConst = 10^9 + 7

countNum :: Int -> Int -> Int
countNum a n = factorial (a + n - 1) `div` (factorial (n - 1) * factorial a)

factorial 0 = 1
factorial n = n * factorial (n - 1)

factors :: Int -> [Int]
factors n = [x | x <- [1..n], n `mod` x == 0]

factorization :: Int -> [Int]
factorization 1 = []
factorization x = v : factorization (x `div` v)
  where
    v = (factors x) !! 1