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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.*;
import java.io.*;
import java.math.*;

	
public class Main {
    
    static class Graph0n {
        private ArrayList<Node0n> dt = new ArrayList<>();
        Graph0n(int sz){
	   	    for(int i=0; i<sz; i++){
	   	        Node0n node1 = new Node0n();
	   	        dt.add(node1);
	   	    }
        }
        public void add(int vn, int val){       dt.get(vn).add(val);                }
        public int get(int vn, int index){      return dt.get(vn).get(index);       }
        public ArrayList<Integer> get(int vn){  return dt.get(vn).getAll();         }
        public int sizeOf(int vn){              return dt.get(vn).size();           }
        public void clear(){    for(int i=0; i<dt.size(); i++){ dt.get(i).clear();  }   }
    }
    
    static class Node0n {   //重みなし無向・有向グラフの頂点
        private ArrayList<Integer> next_vs = new ArrayList<>();
        
        public void add(int val){               next_vs.add(val);       }
        public int get(int ad){                 return next_vs.get(ad); }
        public ArrayList<Integer> getAll(){     return next_vs;         }
        public int size(){                      return next_vs.size();  }
        public void clear(){                    next_vs.clear();        }
    }

    static class Edge {
        int from=-1, v2=-1;     long weight;
        public Edge(int vn, long w){
            this.v2 = vn;
            this.weight = w;
        }
        public Edge(int cm, int vn, long w){
            this.from = cm;
            this.v2 = vn;
            this.weight = w;
        }
    }
    static class Edge2 {
        int v2;     long cost1,cost2;
        public Edge2(int vn, long w1, long w2){
            this.v2 = vn;
            this.cost1 = w1;
            this.cost2 = w2;
        }
    }

    static class Comparator_Edge implements Comparator<Edge>{
        public int compare(Edge a, Edge b){
            if(a.weight>b.weight) return 1;
            else if(a.weight<b.weight) return -1;
            else return 0;
        }
    }
    
	public static void main(String[] args) throws Exception {
		FastScanner sc = new FastScanner();
		PrintWriter out = new PrintWriter(System.out);
		
		long n = sc.nexL();
		
		long ans=1; //k=n
	   	for(int i=2; i<=Math.sqrt(n); i++){
	   	    long q=n;
	   	    if(q%i>0) continue;
	   	    while(q%i==0) q/=i;
	   	    if(q%i==1){
	   	        ans++;
	   	    }
	   	}   //a>=1
	   	
	   	long a1b1c1 = 1;
	   	
	   	List<Edge> facted = factorizationP(n-1);
	   	for(int i=0; i<facted.size(); i++){
	   	    a1b1c1 *= (facted.get(i).v2+1);
	   	}
	   	a1b1c1--;   //k==1
		
		ans += a1b1c1;
		out.println(ans);
		out.flush();
	}
	
	private static List<Edge> factorizationP(long n){
	    List<Edge> ans = new ArrayList<>();
	    
	    long quotient = n;
	    long devider = 2;
	    int exponent = 0;
	    
	    while(quotient%devider == 0){
	        quotient /= devider;
	        exponent++;
	    }
	    if(exponent>0) ans.add(new Edge(exponent, devider));
	    
	    devider = 3;    exponent = 0;
	    while(quotient%devider == 0){
	        quotient /= devider;
	        exponent++;
	    }
	    if(exponent>0) ans.add(new Edge(exponent, devider));
	    
	    devider=1;
	    while(devider <= Math.sqrt(n)){
	        devider+=4; exponent=0;
    	    while(quotient%devider == 0){
    	        quotient /= devider;
    	        exponent++;
    	    }
    	    if(exponent>0) ans.add(new Edge(exponent, devider));
    	    
	        devider+=2; exponent=0;
    	    while(quotient%devider == 0){
    	        quotient /= devider;
    	        exponent++;
    	    }
    	    if(exponent>0) ans.add(new Edge(exponent, devider));
	    }
	    if(quotient>1) ans.add(new Edge(1, quotient));
	    return ans;
	}
	
	private static int INF = (int)1e8;
	private static long INFL = (long)1e17;
	private static long e97 = (long)1e9 + 7;
	
	
	private static int abs(int a){  return (a>=0) ? a: -a;  }
	private static long abs(long a){    return (a>=0) ? a: -a;  }
	private static double abs(double a){    return (a>=0) ? a: -a;  }
	
	private static int min(int a, int b){   return (a>b) ? b : a;   }
	private static long min(long a, long b){    return (a>b) ? b : a;   }
	private static double min(double a, double b){    return (a>b) ? b : a;   }
	
	private static int max(int a, int b){   return (a>b) ? a : b;   }
	private static long max(long a, long b){    return (a>b) ? a : b;   }
	private static double max(double a, double b){    return (a>b) ? a : b;   }
	
	private static int minN(int... ins){
		int min = ins[0];
		for(int i=1; i<ins.length; i++){    if(ins[i] < min) min = ins[i];  }
		return min;
	}
	private static int maxN(int... ins){
		int max = ins[0];
		for(int i=1; i<ins.length; i++){    if(ins[i] > max) max = ins[i];  }
		return max;
	}
	private static long minN(long... ins){
		long min = ins[0];
		for(int i=1; i<ins.length; i++){    if(ins[i] < min) min = ins[i];  }
		return min;
	}
	private static long maxN(long... ins){
		long max = ins[0];
		for(int i=1; i<ins.length; i++){    if(ins[i] > max) max = ins[i];  }
		return max;
	}
	
	private static int minExAd(int[] dt, int ad){
	    int min=Integer.MAX_VALUE;
		for(int i=0; i<dt.length; i++){ if((i != ad) && (dt[i] < min)) min = dt[i]; }
		return min;
	}
	private static long minExAd(long[] dt, int ad){
	    long min=Long.MAX_VALUE;
		for(int i=0; i<dt.length; i++){ if((i != ad) && (dt[i] < min)) min = dt[i]; }
		return min;
	}
	private static int minExVal(int[] dt, int ex_val){
	    int min=Integer.MAX_VALUE;
		for(int i=0; i<dt.length; i++){ if((dt[i] != ex_val) && (dt[i] < min)) min = dt[i]; }
		return min;
	}
	private static long minExVal(long[] dt, long ex_val){
	    long min=Long.MAX_VALUE;
		for(int i=0; i<dt.length; i++){ if((dt[i] != ex_val) && (dt[i] < min)) min = dt[i]; }
		return min;
	}
	
	private static boolean same3(long a, long b, long c){
    	if(a!=b) return false;
    	if(b!=c) return false;
    	if(c!=a) return false;
    	return true;
	}
	private static boolean dif3(long a, long b, long c){
    	if(a==b) return false;
    	if(b==c) return false;
    	if(c==a) return false;
    	return true;
	}
	
	private static double hypod(double a, double b){
	    return Math.sqrt(a*a+b*b);
	}
	
	private static int factorial(int n) {
		int ans=1;
		for(int i=n; i>0; i--){ ans*=i; }
		return ans;
	}
	
	private static long lcm(long m, long n){
	    long ans = m/gcd(m,n);
	    ans *= n;
	    return ans;
	}
	
	private static long gcd(long m, long n) {
		if(m < n) return gcd(n, m);
		if(n == 0) return m;
		return gcd(n, m % n);
	}
	
	private static boolean is_prime(long a){
	    if(a==1) return false;
	    for(int i=2; i<=Math.sqrt(a); i++){  if(a%i == 0) return false;  }
	    return true;
	}
	
    private static long modinv(long a, long p) {    //a|p, >1に注意
        long b = p, u = 1, v = 0;
        while (b>0) {
            long t = a / b;
            long pe = a%b;
            a=b;    b=pe;
            pe= u-t*v;
            u=v;    v=pe;
        }
        u %= p; 
        if (u < 0) u += p;
        return u;
    }
    
    private static long[] fac10E97 = null;
    private static long[] finv10E97 = null;
    private static void Cinit(int max_sz, long p){
        fac10E97 = new long[max_sz+1];
        finv10E97 = new long[max_sz+1];
        fac10E97[0]=1;
        finv10E97[0]=1;
        for(int i=1; i<=max_sz; i++){
            fac10E97[i] = (fac10E97[i-1]*i) % p;
        }
        finv10E97[max_sz] = modinv(fac10E97[max_sz], p);
		for(int i=max_sz; i>1; i--){
		    finv10E97[i-1] = (finv10E97[i]*i) % p;
		}
    }
    
    private static long C10e97(int n, int k, long p){
        long ans = fac10E97[n];
        ans *= finv10E97[k];
        ans %= p;
        ans *= finv10E97[n-k];
        ans %=p;
        if(ans<0) return ans+p;
        else return ans;
    }
	
	private static int pow2(int in){            return in*in;   }
	private static long pow2(long in){          return in*in;   }
	private static double pow2(double in){      return in*in;   }
	
	private static int getDigit2(long num){
		long cf = 1;    int d=0;
		while(num >= cf){   d++;    cf = 1<<d;  }
		return d; //numはd桁の数で、2^dより小さい
	}
	
	private static int getDigit10(long num){
		long cf = 1;    int d=0;
		while(num >= cf){   d++;    cf*=10;     }
		return d; //numはd桁の数で、10^dより小さい
	}
	
	private static boolean isINF(int in){
    	if(((long)in*20)>INF) return true;
    	else return false;
	}
	private static boolean isINFL(long in){
    	if((in*10000)>INFL) return true;
    	else return false;
	}
	
	private static long pow10E97(long ob, long soeji, long p){
	    if(ob==0) return 0;
	    if(soeji==0) return 1;
		if(soeji==2) return (ob*ob)%p;
		
		int d = getDigit2(soeji);
		long[] ob_pow_2pow = new long[d];
		
		ob_pow_2pow[0] = ob;
		for(int i=1; i<d; i++){ ob_pow_2pow[i] = (ob_pow_2pow[i-1]*ob_pow_2pow[i-1])%p;   }
		
		long ans=1;
		for(int i=d-1; i>=0; i--){
			if(soeji >= (long)(1<<i)){
				soeji -= (long)(1<<i);
				ans = (ans*ob_pow_2pow[i])%p;
			}
 		}
		return ans;
	}
	
	
	private static int flag(int pos){               return (1<<pos);        }
	private static boolean isFlaged(int bit, int pos){
    	if((bit&(1<<pos)) > 0) return true;
    	else return false;
	}
	private static int deflag(int bit, int pos){    return bit&~(1<<pos);   }
	
	private static void showflag(int bit){
    	for(int i=0; i<getDigit2(bit); i++){
        	if(isFlaged(bit,i)) System.out.print("O");
        	else System.out.print(".");
    	}
	}
	
	public static int biSearch(int[] dt, int target){
		int left=0, right=dt.length-1;
		int mid=-1;
		while(left<=right){
			mid = (right+left)/2;
			if(dt[mid] == target) return mid;
			if(dt[mid] < target) left=mid+1;
			else right=mid-1;
		}
		return -1;
	}
	public static int biSearchMax(long[] dt, long target){
		int left=-1, right=dt.length;
		int mid=-1;
		
		while((right-left)>1){
			mid = left + (right-left)/2;
			if(dt[mid] < target) left=mid;
			else right=mid;
		}
		return left;//target未満の最大のaddress
	}
	public static int biSearchMaxAL(ArrayList<Integer> dt, long target){
		int left=-1, right=dt.size();
		int mid=-1;
		
		while((right-left)>1){
			mid = left + (right-left)/2;
			if(dt.get(mid) < target) left=mid;
			else right=mid;
		}
		return left;//target未満の最大のaddress
	}
	
    private static int get_root_uf(int[] parent, int index){
        if(parent[index] == index) return index;
        int root = get_root_uf(parent, parent[index]);
        parent[index] = root;
        return root;
    }
    
    private static boolean is_same_uf(int[] parent, int x, int y){
        if(get_root_uf(parent,x) == get_root_uf(parent,y)) return true;
        else return false;
    }
    
    private static void unite_uf(int[] parent, int receiver, int attacker){
        parent[get_root_uf(parent,attacker)] = get_root_uf(parent, receiver);
    }
	
    private static int areaSum_half_open(int[][] acum_plus1, int bY_notInclude, int aY_include, int bX_notInclude, int aX_include){
        int ans = acum_plus1[aY_include+1][aX_include+1];
        ans -= acum_plus1[aY_include+1][bX_notInclude+1];
        ans -= acum_plus1[bY_notInclude+1][aX_include+1];
        ans += acum_plus1[bY_notInclude+1][bX_notInclude+1];
        return ans;
    }
    private static long areaSum_close(long[][] acum_plus1, int bY_Include, int aY_include, int bX_Include, int aX_include){
        long ans = acum_plus1[aY_include+1][aX_include+1];
        ans -= acum_plus1[aY_include+1][bX_Include];
        ans -= acum_plus1[bY_Include][aX_include+1];
        ans += acum_plus1[bY_Include][bX_Include];
        return ans;
    }
    
	private static void nextPerm(ArrayList<Integer> former){
		int pivot=-1;
		int ln = former.size();
		for(int i = ln; i>1; i--){
			if(former.get(i-2) < former.get(i-1)){
				pivot = i-2;
				break;
			}
		}
		if(pivot==-1){
    		former = null;
    		return;
		}
		int pivot_fig = former.get(pivot);
		
		int min_pos=pivot+1;
		int min=former.get(min_pos);
		for(int i=ln-1; i > pivot+1; i--){
			if((former.get(i) < min)&&(former.get(i) > pivot_fig)){
				min_pos=i;
				min=former.get(min_pos);
			}
		}
		Collections.swap(former, pivot, min_pos);
		Collections.sort(former.subList(pivot+1, ln));
	}
    private static ArrayList<boolean[]> combi_enum(int ln, boolean[] former, int now_pos, int rest){
        ArrayList<boolean[]> ans = new ArrayList<>();
        boolean[] ans1 = new boolean[former.length+1];
	   	for(int i=0; i<former.length; i++){
	   	    ans1[i] = former[i];
	   	}
        if(ln==(now_pos+1)){
            switch(rest){
                case 0:
                    ans1[former.length] = false;
                    ans.add(ans1);
                    break;
                case 1:
                    ans1[former.length] = true;
                    ans.add(ans1);
                    break;
            }
        }else{
            if(rest == 0){
                ans1[former.length] = false;
                ans.addAll(combi_enum(ln, ans1, now_pos+1, rest));
            }else{
                ans1[former.length] = false;
                ans.addAll(combi_enum(ln, ans1.clone(), now_pos+1, rest));
                ans1[former.length] = true;
                ans.addAll(combi_enum(ln, ans1.clone(), now_pos+1, rest-1));
            }
        }
        return ans;
    }
    
    private static int[] Xdir4 = {1,0,0,-1};
    private static int[] Ydir4 = {0,1,-1,0};
	    
	private	static int[] Xdir8 = {1,1,1,0,0,-1,-1,-1};
	private	static int[] Ydir8 = {1,0,-1,1,-1,1,0,-1};
	
	private static boolean is_in_area(int y, int x, int h, int w){
	    if(y<0) return false;
	    if(x<0) return false;
	    if(y>=h) return false;
	    if(x>=w) return false;
	    return true;
	}
	
	
	static void show2(boolean[][] dt, int lit_x, int lit_y){
		PrintWriter out = new PrintWriter(System.out);
		for(int i=0; i<dt[0].length; i++){
			for(int j=0; j<dt.length; j++){
				if((i==lit_y) && (j==lit_x)) out.print("X");
				else if(dt[j][i]) out.print("O");
				else out.print(".");
			}
			out.println();
		}
		out.flush();
	}
	static void show2(int[][] dt, String cmnt){
		PrintWriter out = new PrintWriter(System.out);
		for(int i=0; i<dt.length; i++){
			for(int j=0; j<dt[0].length; j++){
				out.print(dt[i][j]+",");
			}
			out.println("<-"+cmnt+":"+i);
		}
		out.flush();
	}
	static void show2(long[][] dt, String cmnt){
		PrintWriter out = new PrintWriter(System.out);
		for(int i=0; i<dt.length; i++){
			for(int j=0; j<dt[0].length; j++){
				out.print(dt[i][j]+",");
			}
			out.println("<-"+cmnt+":"+i);
		}
		out.flush();
	}
	static void disp_que(ArrayDeque<Long> dt){		//上手くいかなかった時用
		long a=0;
		while(dt.size()>0){
			a=dt.removeFirst();
			System.out.print(a);
		}
		System.out.println("\n");
	}
	static void disp_list(List dt){		//上手くいかなかった時用
	   	for(int i=0; i<dt.size(); i++){
			System.out.print(dt.get(i)+",");
		}
		System.out.println("\n");
	}
	
	
	
	
	private static void fill(boolean[] ob, boolean res){
		for(int i=0; i<ob.length; i++){ ob[i] = res;    }
	}
	private static void fill(int[] ob, int res){
		for(int i=0; i<ob.length; i++){ ob[i] = res;    }
	}
	private static void fill(long[] ob, long res){
		for(int i=0; i<ob.length; i++){ ob[i] = res;    }
	}
	private static void fill(char[] ob, char res){
		for(int i=0; i<ob.length; i++){ ob[i] = res;    }
	}
	private static void fill(double[] ob, double res){
		for(int i=0; i<ob.length; i++){ ob[i] = res;    }
	}
	private static void fill(boolean[][] ob, boolean res){
		for(int i=0; i<ob.length; i++){ for(int j=0; j<ob[0].length; j++){  ob[i][j] = res; }   }
	}
	private static void fill(int[][] ob, int res){
		for(int i=0; i<ob.length; i++){ for(int j=0; j<ob[0].length; j++){  ob[i][j] = res; }   }
	}
	private static void fill(long[][] ob, long res){
		for(int i=0; i<ob.length; i++){ for(int j=0; j<ob[0].length; j++){  ob[i][j] = res; }   }
	}
	private static void fill(char[][] ob, char res){
		for(int i=0; i<ob.length; i++){ for(int j=0; j<ob[0].length; j++){  ob[i][j] = res; }   }
	}
	private static void fill(double[][] ob, double res){
		for(int i=0; i<ob.length; i++){ for(int j=0; j<ob[0].length; j++){  ob[i][j] = res; }   }
	}
	private static void fill(int[][][] ob, int res){
		for(int i=0; i<ob.length; i++){	for(int j=0; j<ob[0].length; j++){
				for(int k=0; k<ob[0][0].length; k++){   ob[i][j][k] = res;  }   }   }
	}
	private static void fill(long[][][][] ob, long res){
		for(int i=0; i<ob.length; i++){	for(int j=0; j<ob[0].length; j++){
				for(int k=0; k<ob[0][0].length; k++){	for(int l=0; l<ob[0][0][0].length; l++){
    				    ob[i][j][k][l]=res; }   }   }   }
	}
	
    
static class FastScanner {
	private final InputStream in = System.in;
	private final byte[] buffer = new byte[1024];
	private int ptr = 0;
	private int buflen = 0;
	private boolean hasNextByte() {
		if (ptr < buflen) {
			return true;
		}else{
			ptr = 0;
			try {
				buflen = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			if (buflen <= 0) {
				return false;
			}
		}
		return true;
	}
	
	private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
	private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
	public boolean hasNext() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++; return hasNextByte();}
	public String next() {
		if (!hasNext()) throw new NoSuchElementException();
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while(isPrintableChar(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	public long nexL() {
		if (!hasNext()) throw new NoSuchElementException();
		long n = 0;
		boolean minus = false;
		int b = readByte();
		if (b == '-') {
			minus = true;
			b = readByte();
		}
		if (b < '0' || '9' < b) {
			throw new NumberFormatException();
		}
		while(true){
			if ('0' <= b && b <= '9') {
				n *= 10;
				n += b - '0';
			}else if(b == -1 || !isPrintableChar(b) || b == ':'){
				return minus ? -n : n;
			}else{
				throw new NumberFormatException();
			}
			b = readByte();
		}
	}
	public int nexI() {
		long nl = nexL();
		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
		return (int) nl;
	}
	public double nexD() { return Double.parseDouble(next());}
	
	public void ni(long[] array2){
	   	for(int i=0; i<array2.length; i++){
	   	    array2[i] = nexL();
	   	}
	   	return;
	}
	public void ni(int[] array2){
	   	for(int i=0; i<array2.length; i++){
	   	    array2[i] = nexI();
	   	}
	   	return;
	}
}
}