Numbers with exactly 3 divisors

Difficulty Level : Medium
Last Updated : 11 Jan, 2022

Given a number N, print all numbers in the range from 1 to N having exactly 3 divisors.

Examples:

Input : N = 16
Output : 4 9
4 and 9 have exactly three divisors.
Divisor

Input : N = 49
Output : 4 9 25 49
4, 9, 25 and 49 have exactly three divisors.

Recommended Practice

3 Divisors

Try It!

After having a close look at the examples mentioned above, you have noticed that all the required numbers are perfect squares and that too are only of primes numbers. The logic behind this is, such numbers can have only three numbers as their divisor and also that includes 1 and that number itself resulting into just a single divisor other than a number, so we can easily conclude that required are those numbers which are squares of prime numbers so that they can have only three divisors (1, the number itself and sqrt(number)). So all primes i, such that i*i is less than equal to N are three-prime numbers.

Note: We can generate all primes within a set using any sieve method efficiently and then we should all primes i, such that i*i <=N.

Below is the implementation of the above approach:

C++

// C++ program to print all 
// three-primes smaller than
// or equal to n using Sieve 
// of Eratosthenes
#include <bits/stdc++.h>
using namespace std;
 
// Generates all primes upto n and 
// prints their squares
void numbersWith3Divisors(int n)
{
    bool prime[n+1];
    memset(prime, true, sizeof(prime));
    prime[0] = prime[1] = 0;
 
    for (int p = 2; p*p <= n; p++)
    {
        // If prime[p] is not changed, 
        // then it is a prime
        if (prime[p] == true)
        {
           // Update all multiples of p
           for (int i = p*2; i <= n; i += p)
              prime[i] = false;
        }
    }
 
    // print squares of primes upto n.
    cout << "Numbers with 3 divisors :\n";
    for (int i=0;  i*i <= n ; i++)
        if (prime[i])
          cout << i*i << " ";
}
 
// Driver program 
int main()
{
    // sieve();
    int n = 96;
    numbersWith3Divisors(n); 
 
    return 0;
}

Java

// Java program to print all 
// three-primes smaller than
// or equal to n using Sieve 
// of Eratosthenes
import java.io.*;
import java.util.*;
 
class GFG 
{
     
    // Generates all primes upto n 
    // and prints their squares
    static void numbersWith3Divisors(int n)
    {
        boolean[] prime = new boolean[n+1];
        Arrays.fill(prime, true);
        prime[0] = prime[1] = false;
  
        for (int p = 2; p*p <= n; p++)
        {
             
            // If prime[p] is not changed, 
            // then it is a prime
            if (prime[p] == true)
            {
                // Update all multiples of p
                for (int i=p*2; i<=n; i += p)
                    prime[i] = false;
            }
        }
  
        // print squares of primes upto n
        System.out.println("Numbers with 3 divisors : ");
        for (int i=0;  i*i <= n ; i++)
            if (prime[i])
                System.out.print(i*i + " ");
    }
     
    // Driver program
    public static void main (String[] args) 
    {
        int n = 96;
        numbersWith3Divisors(n); 
    }
}
 
// Contributed by Pramod Kumar

Python3

# Python3 program to print 
# all three-primes smaller than 
# or equal to n using Sieve 
# of Eratosthenes 
 
# Generates all primes upto n 
# and prints their squares 
def numbersWith3Divisors(n): 
  
    prime=[True]*(n+1); 
    prime[0] = prime[1] = False;
    p=2;
    while (p*p <= n):
         
        # If prime[p] is not changed, 
        # then it is a prime 
        if (prime[p] == True):
             
            # Update all multiples of p
            for i in range(p*2,n+1,p):
                prime[i] = False; 
        p+=1; 
 
    # print squares of primes upto n. 
    print("Numbers with 3 divisors :");
    i=0;
    while (i*i <= n): 
        if (prime[i]):
            print(i*i,end=" ");
        i+=1;
 
# Driver program 
 
n = 96; 
numbersWith3Divisors(n); 
 
# This code is contributed by mits

C#

// C# program to print all 
// three-primes smaller than
// or equal to n using Sieve 
// of Eratosthenes
 
class GFG 
{
     
    // Generates all primes upto n 
    // and prints their squares
    static void numbersWith3Divisors(int n)
    {
        bool[] prime = new bool[n+1];
        prime[0] = prime[1] = true;
 
        for (int p = 2; p*p <= n; p++)
        {
             
            // If prime[p] is not changed, 
            // then it is a prime
            if (prime[p] == false)
            {
                // Update all multiples of p
                for (int i = p*2; i <= n; i += p)
                    prime[i] = true;
            }
        }
 
        // print squares of primes upto n
        System.Console.WriteLine("Numbers with 3 divisors : ");
        for (int i=0; i*i <= n ; i++)
            if (!prime[i])
                System.Console.Write(i*i + " ");
    }
     
    // Driver program
    public static void Main() 
    {
        int n = 96;
        numbersWith3Divisors(n); 
    }
}
 
// This code is Contributed by mits

PHP

<?php
// PHP program to print all three-primes
// smaller than or equal to n using Sieve 
// of Eratosthenes 
 
// Generates all primes upto n and 
// prints their squares 
function numbersWith3Divisors($n) 
{ 
    $prime = array_fill(0, $n + 1, true); 
    $prime[0] = $prime[1] = false; 
 
    for ($p = 2; $p * $p <= $n; $p++) 
    { 
        // If prime[p] is not changed, 
        // then it is a prime 
        if ($prime[$p] == true) 
        { 
        // Update all multiples of p 
        for ($i = $p * 2; $i <= $n; $i += $p) 
            $prime[$i] = false; 
        } 
    } 
 
    // print squares of primes upto n. 
    echo "Numbers with 3 divisors :\n"; 
    for ($i = 0; $i * $i <= $n ; $i++) 
        if ($prime[$i]) 
        echo $i * $i . " "; 
} 
 
// Driver Code
$n = 96; 
numbersWith3Divisors($n); 
 
// This code is contributed by mits
?>

Javascript

<script>
    // Javascript program to print all
    // three-primes smaller than
    // or equal to n using Sieve
    // of Eratosthenes
     
    // Generates all primes upto n and
    // prints their squares
    function numbersWith3Divisors(n)
    {
        let prime = new Array(n+1);
        prime.fill(true);
        prime[0] = prime[1] = 0;
 
        for (let p = 2; p*p <= n; p++)
        {
            // If prime[p] is not changed,
            // then it is a prime
            if (prime[p] == true)
            {
               // Update all multiples of p
               for (let i = p*2; i <= n; i += p)
                  prime[i] = false;
            }
        }
 
        // print squares of primes upto n.
        document.write("Numbers with 3 divisors :" + "</br>");
        for (let i = 0;  i*i <= n ; i++)
            if (prime[i])
              document.write(i*i + " ");
    }
 
    // sieve();
    let n = 96;
    numbersWith3Divisors(n);
     
     // This code is contributed by mukesh07.
</script>

Output:

Numbers with 3 divisors :
4 9 25 49

Another Approach:

To print all numbers in the range from 1 to N having exactly 3 divisors, the main calculation is to find those which are squares of prime numbers and less than or equal to the number. We can do this as follows:

Start a loop for integer i from 2 to n.
Check if i is prime or not, which can be done easily using the isPrime(n) method.
If i is prime, check if its square is less than or equal to the given number. This will be checked only for squares of prime numbers, therefore reducing the number of checks.
If the above condition is satisfied, the number will be printed and the loop will continue till i<=n.

In this way, no extra space will be required to store anything.

Here is an implementation of the above logic without using extra space;

C++

// C++ program to print all
// three-primes smaller than
// or equal to n without using
// extra space
#include <bits/stdc++.h>
using namespace std;
 
void numbersWith3Divisors(int);
bool isPrime(int);
 
// Generates all primes upto n and
// prints their squares
void numbersWith3Divisors(int n)
{
    cout << "Numbers with 3 divisors : "
         << endl;
 
    for(int i = 2; i * i <= n; i++)
    {
         
        // Check prime
        if (isPrime(i))
        {
            if (i * i <= n)
            {
                 
                // Print numbers in
                // the order of
                // occurrence
                cout << i * i << " ";
            }
        }
    }
}
 
// Check if a number is prime or not
bool isPrime(int n)
{
    for(int i = 2; i * i <= n; i++)
    {
        if (n % i == 0)
            return false;
    }
    return true;
}
 
// Driver code
int main()
{
    int n = 122;
 
    numbersWith3Divisors(n);
 
    return 0;
}
 
// This code is contributed by vishu2908

Java

// Java program to print all
// three-primes smaller than
// or equal to n without using
// extra space
import java.util.*;
 
class GFG 
{
 
    // 3 divisor logic implementation
    // check if a number is 
    // prime or not
    // if it is a prime then 
    // check if its square
    // is less than or equal to 
    // the given number
    static void numbersWith3Divisors(int n)
    {
        System.out.println("Numbers with 3 divisors : ");
 
        for (int i = 2; i * i <= n; i++) 
        {
             
            // Check prime
            if (isPrime(i)) 
            {
                if (i * i <= n) 
                {
                     
                    // Print numbers in 
                    // the order of
                    // occurrence
                    System.out.print(i * i + " ");
                }
            }
        }
    }
                            
    // Check if a number is prime or not
    public static boolean isPrime(int n)
    {
        for (int i = 2; i * i <= n; i++) 
        {
            if (n % i == 0)
                return false;
        }
        return true;
    }
 
    // Driver program
    public static void main(String[] args)
    {
        int n = 122;
        numbersWith3Divisors(n);
    }
}
 
// Contributed by Parag Pallav Singh

Python3

# Python3 program to print all
# three-primes smaller than
# or equal to n without using
# extra space
 
# 3 divisor logic implementation
# check if a number is  prime or 
# not if it is a prime then check 
# if its square is less than or
# equal to the given number
def numbersWith3Divisors(n):
 
    print("Numbers with 3 divisors : ")
     
    i = 2
    while i * i <= n:
         
        # Check prime
        if (isPrime(i)): 
            if (i * i <= n):
             
                # Print numbers in the order
                # of occurrence
                print(i * i, end = " ")
                 
        i += 1
       
# Check if a number is prime or not
def isPrime(n):
     
    i = 2
    while i * i <= n:
        if n % i == 0:
            return False
             
        i += 1
 
    return True
 
# Driver code
n = 122
 
numbersWith3Divisors(n)
 
# This code is contributed by divyesh072019

C#

// C# program to print all
// three-primes smaller than
// or equal to n without using
// extra space
using System;
 
class GFG{
     
// 3 divisor logic implementation
// check if a number is prime or 
// not if it is a prime then check
// if its square is less than or
// equal to the given number
static void numbersWith3Divisors(int n)
{
    Console.WriteLine("Numbers with 3 divisors : ");
 
    for(int i = 2; i * i <= n; i++) 
    {
         
        // Check prime
        if (isPrime(i)) 
        {
            if (i * i <= n) 
            {
                 
                // Print numbers in the order
                // of occurrence
                Console.Write(i * i + " ");
            }
        }
    }
}
                         
// Check if a number is prime or not
public static bool isPrime(int n)
{
    for(int i = 2; i * i <= n; i++) 
    {
        if (n % i == 0)
            return false;
    }
    return true;
} 
 
// Driver code
static void Main() 
{
    int n = 122;
     
    numbersWith3Divisors(n);
}
}
 
// This code is contributed by divyeshrabadiya07

Javascript

<script>
    // Javascript program to print all
    // three-primes smaller than
    // or equal to n without using
    // extra space
     
      // 3 divisor logic implementation
    // check if a number is prime or
    // not if it is a prime then check
    // if its square is less than or
    // equal to the given number
    function numbersWith3Divisors(n)
    {
        document.write("Numbers with 3 divisors : ");
 
        for(let i = 2; i * i <= n; i++)
        {
 
            // Check prime
            if (isPrime(i))
            {
                if (i * i <= n)
                {
 
                    // Print numbers in the order
                    // of occurrence
                    document.write(i * i + " ");
                }
            }
        }
    }
 
    // Check if a number is prime or not
    function isPrime(n)
    {
        if (n == 0 || n == 1)
            return false;
 
        for(let i = 2; i * i <= n; i++)
        {
            if (n % i == 0)
                return false;
        }
        return true;
    }
     
    let n = 122;
      
    numbersWith3Divisors(n);
 
// This code is contributed by suresh07.
</script>

Output:

Numbers with 3 divisors :
4 9 25 49 121

This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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Numbers with exactly 3 divisors

C++

Java

Python3

C#

PHP

Javascript

C++

Java

Python3

C#

Javascript

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