Problem:
Score : 600 points

Problem Statement

Given is a positive integer N.

We will choose an integer K between 2 and N (inclusive), then we will repeat the operation below until N becomes less than K.

Operation: if K divides N, replace N with N/K; otherwise, replace N with N-K.

In how many choices of K will N become 1 in the end?

Constraints

2 \leq N \leq 10^{12}

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of choices of K in which N becomes 1 in the end.

Sample Input 1

6

Sample Output 1

3

There are three choices of K in which N becomes 1 in the end: 2, 5, and 6.

In each of these choices, N will change as follows:

When K=2: 6 \to 3 \to 1

When K=5: 6 \to 1

When K=6: 6 \to 1

Sample Input 2

3141

Sample Output 2

13

Sample Input 3

314159265358

Sample Output 3

9

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

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace q6
{
    class Program
    {
        static IEnumerable<long> Div(long n)
        {
            var x = 2L;
            yield return 1;
            while (true)
            {
                if ((x * x) > n)
                    break;
                if (n % x == 0)
                {
                    yield return x;
                    if ((n / x) != x)
                    {
                        yield return n/x;
                    }
                }
                x += 1;
            }
            yield return n;
        }

        static void Main(string[] args)
        {
            var N = Convert.ToInt64(Console.ReadLine());
            var cnt = 0;
            var set = new HashSet<long>();
            set.Add(1);
            foreach (var k in Div(N).Skip(1))
            {
                //Console.WriteLine($"d1 {k}");
                var n = N;
                while (n > k && n % k == 0)
                {
                    n = n / k;
                }
                if (n > k)
                    n = n % k;
                if (n == 1 || n == k)
                {
                    //Console.WriteLine($" d1 hit {k}");
                    set.Add(k);
                }
            }
            foreach (var k in Div(N - 1).Skip(1))
            {
                //Console.WriteLine($"  d2 {k}");
                //Console.WriteLine($"  d2 hit {k}");
                set.Add(k);
            }
            set.Remove(0);
            set.Remove(-1);
            set.Remove(1);
            /*
            foreach (var v in set.OrderBy(x => x))
            {
                Console.WriteLine($"res {v}");
            }
            */
            Console.WriteLine(set.Count);
        }
    }
}