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 sys,queue,math,copy,itertools,bisect,collections,heapq

def main():
    NI = lambda : int(sys.stdin.readline())

    N = NI()
    max_i = int(math.log2(N))
    div = []
    for i in range(1,int(math.sqrt(N))+1):
        if N % i == 0:
            div.append((i,N//i)); div.append((N//i,i))

    ans = 0
    for a,b in div:
        if a == 1:
            for i in range(1,int(math.sqrt(b-1))+1):
                if (b-1) % i == 0:
                    p = 2 - (i*i == b-1) - (i == 1)
                    ans += p
        else:
            for i in range(1,max_i+1):
                c = round(math.pow(a,1/i))
                if c ** i == a:
                    if b % c == 1: ans += 1
    print(ans)

if __name__ == '__main__':
    main()