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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using static Assistant.Input;
using static Assistant.Debug;
using System.Linq;
using Assistant;

namespace ABC110D
{
    class Program
    {
        static int n;
        static void Main(string[] args)
        {
            n = RInt;
            var m = RInt;

            var primes = PrimeNumbers(100000);
            var fact = PrimeFactorization(m, primes.ToList());

            //約数リスト <約数、カウント>
            Dictionary<int, int> divs = new Dictionary<int, int>();
            foreach (var f in fact)
            {
                if (divs.ContainsKey(f)) divs[f]++;
                else divs.Add(f, 1);
            }

            Mlong ans = 1;
            Mlong.nCkInit(100100);
            foreach (var d in divs)
            {
                int b = d.Value;
                ans *= Mlong.nCk(b + n - 1, b);
            }
            if (m == 1) ans = 1;

            Console.WriteLine(ans);
        }


        /// <summary>
        /// 自然数n以下の素数prime列挙 (NloglogN)
        /// </summary>
        /// <param name="n"></param>
        /// <returns></returns>
        public static int[] PrimeNumbers(int n)
        {
            int max = n;
            var searchList = Enumerable.Range(2, max).ToList();

            var pList = new List<int>();
            while (searchList.Count > 0)
            {
                int cand = searchList.First();

                pList.Add(cand);

                searchList.RemoveAll(x => x % cand == 0);
            }

            return pList.ToArray();
        }

        /// <summary>
        /// 自然数nの素因数分解 
        /// </summary>
        /// <param name="n"></param>
        /// <returns></returns>
        public static int[] PrimeFactorization(int n, List<int> pList)
        {
            int N = n;
            //var pList = PrimeNumbers((int)Math.Sqrt(n)).ToList();

            var ansList = new List<int>();
            for (int i = 0; i < pList.Count;)
            {
                if (N % pList[i] == 0)
                {
                    ansList.Add(pList[i]);
                    N /= pList[i];
                }
                else i++;
            }
            if (ansList.Count == 0 || N != 1) ansList.Add(N);

            return ansList.ToArray();
        }
    }

    /// <summary>
    /// Mod計算型 若干遅いかも
    /// </summary>
    public struct Mlong
    {
        long _v;
        const long mod = 1000000007;
        public Mlong(long n = 0) : this() { _v = n >= mod ? n % mod : n; }

        public static implicit operator Mlong(long _x) => new Mlong(_x);
        public static implicit operator long(Mlong _x) => _x._v;
        public static Mlong operator +(Mlong m1, Mlong m2) { long m = m1._v + m2._v; return m >= mod ? m - mod : m; }
        public static Mlong operator -(Mlong m1, Mlong m2) { long m = m1._v - m2._v; return m >= 0 ? m : m + mod; }
        public static Mlong operator *(Mlong m1, Mlong m2) => m1._v * m2._v % mod;
        public static Mlong operator /(Mlong m1, Mlong m2) => m1._v * ModPow(m2._v, mod - 2) % mod;

        public static long ModPow(long a, long n)
        {
            if (n == 0) return 1;
            else if (n % 2 == 1) return a * ModPow(a, n - 1) % mod;
            else return ModPow(a * a, n / 2);
        }

        static Mlong[] fac, finv, inv;
        public static void nCkInit(int max)
        {
            fac = new Mlong[max]; finv = new Mlong[max]; inv = new Mlong[max];
            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (int i = 2; i < max; i++)
            {
                fac[i] = fac[i - 1] * i;
                inv[i] = mod - inv[mod % i] * (mod / i);
                finv[i] = finv[i - 1] * inv[i];
            }
        }

        public static Mlong nCk(int n, int k)
        {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * finv[k] * finv[n - k];
        }
    }

}

namespace Assistant
{
    static class Input
    {
        static List<string> line = new List<string>();
        static int index = 0;
        static String RNext()
        {
            if (line.Count <= index) line.AddRange(Console.ReadLine().Split());
            return line[index++];
        }
        public static int RInt => int.Parse(RNext());
        public static long RLong => long.Parse(RNext());
        public static int[] RInts => Console.ReadLine().Split().Select(int.Parse).ToArray();
        public static long[] RLongs => Console.ReadLine().Split().Select(long.Parse).ToArray();
        public static string RString => RNext();
        //以下未テスト
        public static int[] RIntsC(int c) => Enumerable.Repeat(0, c).Select(x => int.Parse(RNext())).ToArray();
        public static long[] RLongsC(int c) => Enumerable.Repeat(0, c).Select(x => long.Parse(RNext())).ToArray();
        public static char[][] RMap(int h) => Enumerable.Repeat(0, h).Select(x => Console.ReadLine().ToCharArray()).ToArray();
    }

    static class Debug
    {
        static public void Draw2D<T>(T[,] map, int mode = 1)
        {
#if DEBUG
            int W = map.GetLength(0);
            int H = map.GetLength(1);

            string[,] map2 = new string[W + 1, H + 1];
            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (i == 0 && j == 0) map2[i, j] = 0.ToString();
                    else if (i == 0) map2[i, j] = (j - 1).ToString();
                    else if (j == 0) map2[i, j] = (i - 1).ToString();
                    else map2[i, j] = map[i - 1, j - 1].ToString();
                }
            }

            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (mode == 0) Console.Write(map2[i, j].Last());
                    if (mode == 1) Console.Write(map2[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
#endif
        }

        public static void Draw1D<T>(T[] array, int mode = 0)
        {
#if DEBUG
            Console.WriteLine(string.Join(" ", array));
#endif
        }
    }
}