Given a number n, count total perfect divisors of n. Perfect divisors are those divisors which are square of some integer. For example a perfect divisor of 8 is 4.
Examples:
Input : n = 16 Output : 3 Explanation : There are only 5 divisor of 16: 1, 2, 4, 8, 16. Only three of them are perfect squares: 1, 4, 16. Therefore the answer is 3 Input : n = 7 Output : 1
Naive approach
A brute force is find all the divisors of a number. Count all divisors that are perfect squares.
C++
// Below is C++ code to count total perfect Divisors #include<bits/stdc++.h> using namespace std; // Utility function to check perfect square number bool isPerfectSquare(int n) { int sq = (int) sqrt(n); return (n == sq * sq); } // Returns count all perfect divisors of n int countPerfectDivisors(int n) { // Initialize result int count = 0; // Consider every number that can be a divisor // of n for (int i=1; i*i <= n; ++i) { // If i is a divisor if (n%i == 0) { if (isPerfectSquare(i)) ++count; if (n/i != i && isPerfectSquare(n/i)) ++count; } } return count; } // Driver code int main() { int n = 16; cout << "Total perfect divisors of " << n << " = " << countPerfectDivisors(n) << "\n"; n = 12; cout << "Total perfect divisors of " << n << " = " << countPerfectDivisors(n); return 0; } |
Java
// Java code to count // total perfect Divisors import java.io.*; class GFG { // Utility function to check // perfect square number static boolean isPerfectSquare(int n) { int sq = (int) Math.sqrt(n); return (n == sq * sq); } // Returns count all // perfect divisors of n static int countPerfectDivisors(int n) { // Initialize result int count = 0; // Consider every number // that can be a divisor of n for (int i = 1; i * i <= n; ++i) { // If i is a divisor if (n % i == 0) { if (isPerfectSquare(i)) ++count; if (n / i != i && isPerfectSquare(n / i)) ++count; } } return count; } // Driver code public static void main (String[] args) { int n = 16; System.out.print("Total perfect " + "divisors of " + n); System.out.println(" = " + countPerfectDivisors(n)); n = 12; System.out.print("Total perfect " + "divisors of " + n); System.out.println(" = " + countPerfectDivisors(n)); } } // This code is contributed by ajit |
Python3
# Python3 implementation of Naive method # to count all perfect divisors import math def isPerfectSquare(x) : sq = (int)(math.sqrt(x)) return (x == sq * sq) # function to count all perfect divisors def countPerfectDivisors(n) : # Initialize result cnt = 0 # Consider every number that # can be a divisor of n for i in range(1, (int)(math.sqrt(n)) + 1) : # If i is a divisor if ( n % i == 0 ) : if isPerfectSquare(i): cnt = cnt + 1 if n/i != i and isPerfectSquare(n/i): cnt = cnt + 1 return cnt # Driver program to test above function print("Total perfect divisor of 16 = ", countPerfectDivisors(16)) print("Total perfect divisor of 12 = ", countPerfectDivisors(12)) # This code is contributed by Saloni Gupta |
C#
// C# code to count // total perfect Divisors using System; class GFG { // Utility function to check // perfect square number static bool isPerfectSquare(int n) { int sq = (int) Math.Sqrt(n); return (n == sq * sq); } // Returns count all // perfect divisors of n static int countPerfectDivisors(int n) { // Initialize result int count = 0; // Consider every number // that can be a divisor of n for (int i = 1; i * i <= n; ++i) { // If i is a divisor if (n % i == 0) { if (isPerfectSquare(i)) ++count; if (n / i != i && isPerfectSquare(n / i)) ++count; } } return count; } // Driver code static public void Main () { int n = 16; Console.Write("Total perfect " + "divisors of " + n); Console.WriteLine(" = " + countPerfectDivisors(n)); n = 12; Console.Write("Total perfect " + "divisors of " + n); Console.WriteLine(" = " + countPerfectDivisors(n)); } } // This code is contributed // by akt_mit |
PHP
<?php // PHP code to count // total perfect Divisors // function to check // perfect square number function isPerfectSquare($n) { $sq = sqrt($n); return ($n == $sq * $sq); } // Returns count all // perfect divisors of n function countPerfectDivisors($n) { // Initialize result $count = 0; // Consider every number // that can be a divisor // of n for ($i = 1; $i * $i <= $n; ++$i) { // If i is a divisor if ($n % $i == 0) { if (isPerfectSquare($i)) ++$count; if ($n / $i != $i && isPerfectSquare($n / $i)) ++$count; } } return $count; } // Driver Code $n = 16; echo "Total perfect divisors of ", $n, " = ", countPerfectDivisors($n), "\n"; $n = 12; echo "Total perfect divisors of ", $n, " = ", countPerfectDivisors($n); // This code is contributed by ajit ?> |
Javascript
<script> // JavaScript program for the above approach // Utility function to check // perfect square number function isPerfectSquare(n) { let sq = Math.sqrt(n); return (n == sq * sq); } // Returns count all // perfect divisors of n function countPerfectDivisors(n) { // Initialize result let count = 0; // Consider every number // that can be a divisor of n for (let i = 1; i * i <= n; ++i) { // If i is a divisor if (n % i == 0) { if (isPerfectSquare(i)) ++count; if (n / i != i && isPerfectSquare(n / i)) ++count; } } return count; } // Driver Code let n = 16; document.write("Total perfect " + "divisors of " + n); document.write(" = " + countPerfectDivisors(n) + "<br/>"); n = 12; document.write("Total perfect " + "divisors of " + n); document.write(" = " + countPerfectDivisors(n)); // This code is contributed by chinmoy1997pal. </script> |
Output: Total Perfect divisors of 16 = 3 Total Perfect divisors of 12 = 2
Time complexity: O(sqrt(n))
Auxiliary space: O(1)
Efficient approach (For large number of queries)
The idea is based on Sieve of Eratosthenes. This approach is beneficial if there are large number of queries. Following is the algorithm to find all perfect divisors up to n numbers.
- Create a list of n consecutive integers from 1 to n:(1, 2, 3, …, n)
- Initially, let d be 2, the first divisor
- Starting from 4(square of 2) increment all the multiples of 4 by 1 in perfectDiv[] array, as all these multiples contain 4 as perfect divisor. These numbers will be 4d, 8d, 12d, … etc
- Repeat the 3rd step for all other numbers. The final array of perfectDiv[] will contain all the count of perfect divisors from 1 to n
Below is implementation of above steps.
C++
// Below is C++ code to count total perfect // divisors #include<bits/stdc++.h> using namespace std; #define MAX 100001 int perfectDiv[MAX]; // Pre-compute counts of all perfect divisors // of all numbers upto MAX. void precomputeCounts() { for (int i=1; i*i < MAX; ++i) { // Iterate through all the multiples of i*i for (int j=i*i; j < MAX; j += i*i) // Increment all such multiples by 1 ++perfectDiv[j]; } } // Returns count of perfect divisors of n. int countPerfectDivisors(int n) { return perfectDiv[n]; } // Driver code int main() { precomputeCounts(); int n = 16; cout << "Total perfect divisors of " << n << " = " << countPerfectDivisors(n) << "\n"; n = 12; cout << "Total perfect divisors of " << n << " = " << countPerfectDivisors(n); return 0; } |
Java
// Java code to count total perfect // divisors class GFG { static int MAX = 100001; static int[] perfectDiv = new int[MAX]; // Pre-compute counts of all perfect divisors // of all numbers upto MAX. static void precomputeCounts() { for (int i = 1; i * i < MAX; ++i) { // Iterate through all the multiples of i*i for (int j = i * i; j < MAX; j += i * i) // Increment all such multiples by 1 ++perfectDiv[j]; } } // Returns count of perfect divisors of n. static int countPerfectDivisors(int n) { return perfectDiv[n]; } // Driver code public static void main (String[] args) { precomputeCounts(); int n = 16; System.out.println("Total perfect divisors of " + n + " = " + countPerfectDivisors(n)); n = 12; System.out.println("Total perfect divisors of " + n + " = " + countPerfectDivisors(n)); } } // This code is contributed by mits |
Python3
# Below is Python3 code to count total perfect # divisors MAX = 100001 perfectDiv= [0]*MAX # Pre-compute counts of all perfect divisors # of all numbers upto MAX. def precomputeCounts(): i=1 while i*i < MAX: # Iterate through all the multiples of i*i for j in range(i*i,MAX,i*i): # Increment all such multiples by 1 perfectDiv[j] += 1 i += 1 # Returns count of perfect divisors of n. def countPerfectDivisors( n): return perfectDiv[n] # Driver code if __name__ == "__main__": precomputeCounts() n = 16 print ("Total perfect divisors of " , n , " = " ,countPerfectDivisors(n)) n = 12 print ( "Total perfect divisors of " ,n ," = " ,countPerfectDivisors(n)) |
C#
// C# code to count total perfect // divisors using System; class GFG { static int MAX = 100001; static int[] perfectDiv = new int[MAX]; // Pre-compute counts of all perfect // divisors of all numbers upto MAX. static void precomputeCounts() { for (int i = 1; i * i < MAX; ++i) { // Iterate through all the multiples of i*i for (int j = i * i; j < MAX; j += i * i) // Increment all such multiples by 1 ++perfectDiv[j]; } } // Returns count of perfect divisors of n. static int countPerfectDivisors(int n) { return perfectDiv[n]; } // Driver code public static void Main() { precomputeCounts(); int n = 16; Console.WriteLine("Total perfect divisors of " + n + " = " + countPerfectDivisors(n)); n = 12; Console.WriteLine("Total perfect divisors of " + n + " = " + countPerfectDivisors(n)); } } // This code is contributed by mits |
PHP
<?php // Below is PHP code to count total // perfect divisors $MAX = 10001; $perfectDiv = array_fill(0, $MAX, 0); // Pre-compute counts of all perfect // divisors of all numbers upto MAX. function precomputeCounts() { global $MAX, $perfectDiv; for ($i = 1; $i * $i < $MAX; ++$i) { // Iterate through all the multiples // of i*i for ($j = $i * $i; $j < $MAX; $j += $i * $i) // Increment all such multiples by 1 ++$perfectDiv[$j]; } } // Returns count of perfect divisors of n. function countPerfectDivisors($n) { global $perfectDiv; return $perfectDiv[$n]; } // Driver code precomputeCounts(); $n = 16; echo "Total perfect divisors of " . $n . " = " . countPerfectDivisors($n) . "\n"; $n = 12; echo "Total perfect divisors of " . $n . " = " . countPerfectDivisors($n); // This code is contributed by mits ?> |
Javascript
<script> // Javascript code to count total perfect // divisors let MAX = 100001;; let perfectDiv = new Array(MAX); for(let i = 0; i < MAX; i++) { perfectDiv[i] = 0; } // Pre-compute counts of all perfect divisors // of all numbers upto MAX. function precomputeCounts() { for(let i = 1; i * i < MAX; ++i) { // Iterate through all the multiples of i*i for(let j = i * i; j < MAX; j += i * i) // Increment all such multiples by 1 ++perfectDiv[j]; } } // Returns count of perfect divisors of n. function countPerfectDivisors(n) { return perfectDiv[n]; } // Driver code precomputeCounts(); let n = 16; document.write("Total perfect divisors of " + n + " = " + countPerfectDivisors(n) + "<br>"); n = 12; document.write("Total perfect divisors of " + n + " = " + countPerfectDivisors(n)); // This code is contributed by rag2127 </script> |
Output:
Total Perfect divisors of 16 = 3 Total Perfect divisors of 12 = 2
Time complexity: O(MAX * log(log (MAX)))
Auxiliary space: O(MAX)
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