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 Ruby (2.3.3) (judge verdict: Wrong Answer)

N,M = gets.split.map(&:to_i)
MOD = 10**9 + 7

#dp[i][j]: a1*a2*...ai=jとなるaの場合の数
#求めるものはdp[M][N]
dp = Hash.new{|hash,value| hash[value] = Array.new(N+1, 0)}

#Mの約数列挙
divs =[]
for i in 1..Math.sqrt(M) do
  if M%i == 0
    divs << i
    if i != Math.sqrt(M)
      divs << M/i
    end
  end
end

divs.each do |d|
  dp[d][1] = 1
end

for j in 2..N do
  divs.each do |d1|
    divs.each do |d2|
      if d1*d2 <= M
        dp[d1*d2][j] += dp[d1][j-1]
        dp[d1*d2][j] %MOD
      end
    end
  end
end

puts dp[M][N]