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 Lua (5.3.2) (judge verdict: Runtime Error)

local mfl, mce = math.floor, math.ceil
local mod = 1000000007
local function badd(x, y) return (x + y) % mod end
local function bmul(x, y) return (x * y) % mod end

local function getprimes(x)
  local primes = {}
  local allnums = {}
  for i = 1, x do allnums[i] = true end
  for i = 2, x do
    if allnums[i] then
      table.insert(primes, i)
      local lim = mfl(x / i)
      for j = 2, lim do
        allnums[j * i] = false
      end
    end
  end
  return primes
end
local primes = getprimes(31623)

local function modpow(src, pow)
  local res = 1
  while 0 < pow do
    if pow % 2 == 1 then
      res = bmul(res, src)
      pow = pow - 1
    end
    src = bmul(src, src)
    pow = mfl(pow / 2)
  end
  return res
end

local function modinv(src)
  return modpow(src, mod - 2)
end

local fact = {1}
for i = 2, 100100 do
  fact[i] = bmul(fact[i - 1], i)
end

local function getComb(n, k)
  return bmul(fact[n], modinv(bmul(fact[k], fact[n - k])))
end

local n, m = io.read("*n", "*n")
local tm = m
local ret = 1
for i = 1, #primes do
  local p = primes[i]
  if tm < p * p then break end
  local cnt = 0
  while tm % p == 0 do
    tm = tm // p
    cnt = cnt + 1
  end
  if 0 < cnt then
    ret = bmul(ret, getComb(n + cnt - 1, n - 1))
  end
end
if tm ~= 1 then
  ret = bmul(ret, n)
end
print(ret)