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 Python (3.4.3) (judge verdict: Accepted)

def primelist(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]

def factorization(n):
    plist = primelist(int(n**0.5))
    factor = []
    for p in plist:
        k = 0
        while n%p==0:
            k+=1
            n//=p
        if k>0:
            factor.append((p,k))
        if n==1:
            break
    if n!=1:
        factor.append((n,1))

    return factor

def primelist(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]

def factorization(n):
    plist = primelist(int(n**0.5))
    factor = []
    for p in plist:
        k = 0
        while n%p==0:
            k+=1
            n//=p
        if k>0:
            factor.append((p,k))
        if n==1:
            break
    if n!=1:
        factor.append((n,1))

    return factor

def divisors(n):
    divisors = []
    for i in range(1, int(n**0.5)+1):
        if n % i == 0:
            divisors.append(i)
            if i != n // i:
                divisors.append(n//i)

    # divisors.sort()
    return divisors


def main():
    n = int(input())
    f1 = divisors(n)
    f2 = factorization(n-1)
        
    s = 1
    for f_i in f2:
        s *=(f_i[1]+1)

    for d in f1[1:]:
        t = test(n,d)
        if t==1:
            s+=1

    print(s-1)
    

def test(n,k):

    while n>=k:
        if n%k==0:
            n//=k
        else:
            n%=k
    return n



if __name__ == "__main__":
    main()