Numbers with exactly 3 divisors
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.
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
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