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 (2.4.0) (judge verdict: Wrong Answer)

import math
n = int(input())
 
m = n-1
ans1 = 0
for i in range(1, int(math.sqrt(m))):
  if m%i == 0:
    ans1 += 2
if m % int(math.sqrt(m)) == 0:
  ans1 += 1

ans2 = 0
for i in range(2, int(math.sqrt(n))):
  if n%i == 0:
    k = n
    while k % i == 0:
      k = k // i
    if k % i == 1:
      ans2 += 1
    k = n
    j = n/i
    while k % j == 0:
      k = k // j
    if k % j == 1:
      ans2 += 1
      
if n % int(math.sqrt(n)) == 0:
    k = n
    while k % int(math.sqrt(n)) == 0:
      k = k // int(math.sqrt(n))
    if k % int(math.sqrt(n)) == 1:
      ans2 += 1
print(ans1+ans2)