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 PyPy3 (7.3.0) (judge verdict: Accepted)

import math
#import numpy as np
import queue
from collections import deque,defaultdict
import heapq as hpq
from sys import stdin,setrecursionlimit
#from scipy.sparse.csgraph import dijkstra
#from scipy.sparse import csr_matrix
ipt = stdin.readline
setrecursionlimit(10**7)

def main():
    n = int(ipt())
    if n == 2:
        print(1)
        exit()

    def al_div(n):
        pn = 1
        div = []
        while pn**2 < n:
            if n%pn == 0:
                div.append(pn)
                div.append(n//pn)
            pn += 1
        if pn**2 == n:
            div.append(pn)
        return div
    ans = 0
    for i in al_div(n-1):
        if i == 1:
            continue
        else:
            if n%i == 1:
                ans += 1
    for i in al_div(n):
        if i == 1:
            continue
        ni = n
        while ni%i == 0:
            ni //= i
        if ni%i == 1:
            ans += 1

    print(ans)

    return

if __name__ == '__main__':
    main()