Valor mínimo que excede X cuyo recuento de divisores tiene diferente paridad con el recuento de divisores de X

Dado un entero X , la tarea es determinar el valor mínimo de Y mayor que X , tal que el recuento de divisores de X e Y tenga paridades diferentes .

Ejemplos:

Entrada: X = 5
Salida: 9
Explicación: La cuenta de divisores de 5 y 9 son 2 y 3 respectivamente, que son de diferente paridad.

Entrada: X = 9
Salida: 10
Explicación: Las cuentas de los divisores de 9 y 10 son 3 y 4, que son de diferente paridad.

Enfoque ingenuo: el enfoque más simple para resolver el problema es iterar cada número a partir de X + 1 hasta obtener un elemento con un número de divisores con paridad opuesta a la de X.

Complejidad de Tiempo: O((1+√X) ² )
Espacio Auxiliar: O(1)

A continuación se muestra la implementación del enfoque anterior:

// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count divisors of n
int divisorCount(int n)
{
    int x = 0;
    for (int i = 1; i <= sqrt(n); i++) {
        if (n % i == 0) {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
int minvalue_y(int x)
{
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
 
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1)
           == (divisorCount(y) & 1))
        y++;
    return y;
}
 
// Driver Code
int main()
{
    // Given X
    int x = 5;
 
    // Function call
    cout << minvalue_y(x) << endl;
    return 0;
}

// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to count divisors of n
static int divisorCount(int n)
{
    int x = 0;
    for(int i = 1; i <= Math.sqrt(n); i++)
    {
        if (n % i == 0)
        {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
 
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1) == (divisorCount(y) & 1))
        y++;
         
    return y;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given X
    int x = 5;
 
    // Function call
    System.out.println(minvalue_y(x));
}
}
 
// This code is contributed by chitranayal

// C# program for the above approach
using System;
  
class GFG{
  
// Function to count divisors of n
static int divisorCount(int n)
{
    int x = 0;
    for(int i = 1; i <= Math.Sqrt(n); i++)
    {
        if (n % i == 0)
        {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
  
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
      
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
  
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1) == (divisorCount(y) & 1))
        y++;
          
    return y;
}
  
// Driver Code
public static void Main()
{
      
    // Given X
    int x = 5;
  
    // Function call
    Console.WriteLine(minvalue_y(x));
}
}
 
// This code is contributed by susmitakundugoaldanga

# Python program for the above approach
 
# Function to count divisors of n
def divisorCount(n):
    x = 0;
    for i in range(1, n):
        if (n % i == 0):
            if (i == n // i):
                x += 1;
            else:
                x += 2;
        if(i * i > n):
            break;
 
    return x;
 
# Function to find the minimum
# value exceeding x whose count
# of divisors has different parity
# with count of divisors of X
def minvalue_y(x):
   
    # Divisor count of x
    a = divisorCount(x);
    y = x + 1;
 
    # Iterate from x + 1 and
    # check for each element
    while ((a & 1) == (divisorCount(y) & 1)):
        y += 1;
 
    return y;
 
# Driver Code
if __name__ == '__main__':
   
    # Given X
    x = 5;
 
    # Function call
    print(minvalue_y(x));
 
# This code is contributed by 29AjayKumar

<script>
 
// javascript program of the above approach
 
// Function to count divisors of n
function divisorCount(n)
{
    let x = 0;
    for(let i = 1; i <= Math.sqrt(n); i++)
    {
        if (n % i == 0)
        {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
function minvalue_y(x)
{
     
    // Divisor count of x
    let a = divisorCount(x);
    let y = x + 1;
 
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1) == (divisorCount(y) & 1))
        y++;
         
    return y;
}
 
    // Driver Code
     
     // Given X
    let x = 5;
 
    // Function call
    document.write(minvalue_y(x));
 
// This code is contributed by target_2.
</script>

9

Enfoque eficiente: el problema se puede resolver con base en las siguientes observaciones:

• Si X es un cuadrado perfecto , entonces X + 1 será la respuesta.
• De lo contrario, (1 + √X) ² será la respuesta.

Siga los pasos a continuación para resolver el problema:

1. Comprueba si X es un cuadrado perfecto . Si se encuentra que es cierto, imprima X + 1 .
2. De lo contrario, imprime (1 + piso (√X)) ² ) .

A continuación se muestra la implementación del enfoque anterior:

// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
int minvalue_y(int x)
{
    // Check if x is
    // perfect square
    int n = sqrt(x);
    if (n * n == x)
        return x + 1;
 
    return pow(n + 1, 2);
}
 
// Driver Code
int main()
{
    int x = 5;
    cout << minvalue_y(x) << endl;
    return 0;
}

// Java program for the above approach
import java.util.*;
    
class GFG{
    
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Check if x is
    // perfect square
    int n = (int)Math.sqrt(x);
    if (n * n == x)
        return x + 1;
  
    return (int)Math.pow(n + 1, 2);
}
   
// Driver Code
public static void main(String[] args)
{
    int x = 5;
      
    System.out.print(minvalue_y(x));
}
}
 
// This code is contributed by sanjoy_62

# Python3 program for the above approach
 
# Function to find the minimum
# value exceeding x whose count
# of divisors has different parity
# with count of divisors of X
def minvalue_y(x):
     
    # Check if x is
    # perfect square
    n = int(pow(x, 1 / 2))
     
    if (n * n == x):
        return x + 1
         
    return(pow(n + 1, 2))
 
# Driver Code
if __name__ == '__main__':
     
    x = 5
 
    print(minvalue_y(x))
 
# This code is contributed by Rajput-Ji

// C# program for the above approach
using System;
     
class GFG{
     
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Check if x is
    // perfect square
    int n = (int)Math.Sqrt(x);
    if (n * n == x)
        return x + 1;
         
    return (int)Math.Pow(n + 1, 2);
}
    
// Driver Code
public static void Main()
{
    int x = 5;
     
    Console.WriteLine(minvalue_y(x));
}
}
 
// This code is contributed by code_hunt

<script>
 
// JavaScript program to implement
// the above approach
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
function minvalue_y(x)
{
      
    // Check if x is
    // perfect square
    let n = Math.floor(Math.sqrt(x));
    if (n * n == x)
        return x + 1;
   
    return Math.floor(Math.pow(n + 1, 2));
}
 
// Driver code
 
    let x = 5;
       
    document.write(minvalue_y(x));
     
</script>

9

Tiempo Complejidad: O(1)
Espacio Auxiliar: O(1)