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: Accepted)

from collections import defaultdict

N = int(input())
X, Y = N-1, N
dic = defaultdict(int)
for p in range(2, int(N**0.5)+1):
    if X % p == 0:
        cnt = 0
        while X % p == 0:
            X //= p
            cnt += 1
        dic[p] = cnt
    if Y % p == 0:
        cnt = 0
        while Y % p == 0:
            Y //= p
            cnt += 1
        dic[p] = cnt
if X > 1:
    dic[X] = 1
if Y > 1:
    dic[Y] = 1
f = [1]
for p in dic.keys():
    f += [x*pow(p, c) for x in f for c in range(1, dic[p]+1)]
ans = 0
for x in f:
    if x == 1:
        continue
    y = N
    while y % x == 0:
        y //= x
    if y % x == 1:
        ans += 1
print(ans)