C Tutorial

C Tutorial C Language Environment Setup Execution flow of C program C printf and Scanf C Data type C Token Variable in C Operators in C Comments in C Escape Sequence in C C – Storage Classes C Decision control statement Loop Statement in C Break, continue and goto statement in C Type Casting in C Function in C Recursion in C String in C C Array Pointer in C Dynamic memory allocation C –Structure Nested Structure in C Union in C File Handling in C C pre-processor Static Function In C Sizeof In C Selection Sort In C Scope Of Variables In C Runtime Vs Compile Time In C Random Access Lseek In C Queue Implementation In C Pseudo Code In C Prototype In C Pointer To Pointer In C Pointer Arithmetic In C Passing Array To Function In C Null Character In C Merge Sort In C Macros In C Library Functions In C Memory Leak In C Int In C Goto And Labels In C Fibonacci Series In C Fflush In C Derived Data Types In C Data Types In C Const Vs Volatile In C Character Set In C Character Class Tests In C Calloc In C C Pointers Arrays In C Include In C Clrscr In C C Vs Java String Literals In C Types Of Pointers In C Variables In C Volatile In C Why C Is A Middle Level Language Infix To Postfix Program In C Ceil function in C LCM of two numbers in C Quick sort in C Static in C function pointer as argument in C Top Array Keywords in C Add two numbers using the function in C Armstrong program in C using function Array, Declaring Arrays and Array Initialization Limitations of Inline Function in C Merge and Merge sort with example in C Do-While Loop in C For Loop in C While-Loop in C Difference between while and do-while loop in C Array Of Structures in C Data Structures And Algorithms in C Types Of Structures In C How to Avoid Structure Padding in C Use of Structure in C Do WHILE LOOP in C Programming Examples For Loop in C Programming Examples Entry Control Loop in C Exit control loop in C Infinite loop in C Nested loop in C pow() function in C String Handling functions in C Prime Number code in C Factorial Program in C using For Loop Factorial Program in C Using While Loop Fibonacci Series in C Using For Loop Fibonacci series in C using while loop Prime Number Program in C using for Loop While Loop in C programming examples Built-in functions in C Assert() Function C vs Java Strings Call Back Function in Embedded C Else If Ladder fgets() function Ftell() Function getc() function getch() function gets() function Heap Sort Nested if-else statement Pi() Function Positioning of file Write() function abs() function in C Attributes in C C program to find factorial of a number using Recursion Ferror() in c fopen() function in C Fibonacci series program in C using Recursion Formatted Input and output function in C Snake Game in C User Defined Functions in C Beep() function in C Cbrt() function in C Hook() function in C Isalnum() function in C C Program to find the Roots of a Quadratic Equation C Switch Statements Difference between rand() and srand() function in C Difference between while and for loop in C Doubly Linked list in C Example of Iteration in C How to use atoi() function in C How to use floor() function in C How to use sine() function in C How to use Typedef Struct in C Integer Promotions in C C Program Swap Numbers in cyclic order Using Call by Reference C Program to Find Largest Number Using Dynamic Memory Allocation C Program to Find the Largest Number using Ternary Operator C/C++ Program to Find the Size of int, float, double and char Find the Largest Three Distinct Elements in an Array using C/C++ Loop Questions in C Modulus on Negative Numbers in C Multiplication table program in C using For loop Nested Loops in C Programming Examples C Program for Mean and Median of an Unsorted Array Results of Comparison Operations in C and C++ Reverse a Stack using Recursion in C Simple hash() function in C strcat() Function in C Sum of N numbers in C using For loop Use of free() function in C Write a program that produces different results in C and C++ C Function Argument and Return Values Keywords in C Bank management system in C Calendar application in C Floor() Function in C Free() Function in C How to delete a file in C How to move a text in C Remove an element from an array in C Unformatted input() and output() function in C What are linker and loader in C SJF Scheduling Program in C Socket Programming in C Structure in C Tower of Hanoi in C Union Program in C Variable Declaration in C What is Linked List in C While Loop Syntax in C fork() in C GCD program in C Branching Statements in C Comma Operator in C Control statement in C Double Specifier in C How to create a binary file in C Long int in C Palindrome Number in C Pure Virtual Function in C Run Time Polymorphism in C Types of Array in C Types of Function in C What is a buffer in C What is required in each C Program Associativity of Operators in C Bit Stuffing Program in C Actual and Formal Parameters Addition of two Numbers in C Advantages of function in C Arithmetic Progression Program in C Binomial Coefficient Program in C Difference between Array and List in C Diffie-Hellman Algorithm in C How to convert a number to words in C How to convert a string to hexadecimal in C Difference between If and Switch Statement in C C and C++ Binary Files C program that does not Suspend when Ctrl+Z is Pressed Different ways to Declare the Variable as Constant in C Range of Int in C C Program to find the size of a File FIFO Example in the C Language For loop in C Programming GCD program of two numbers in C GPA Calculator in C How to Calculate Time Complexity in C How to include graphics.h in C How to measure time taken by a function in C How to return a Pointer from a Function in C What is the main in C Addition of Matrix in C Booleans in C C Program for Extended Euclidean algorithms C Program of Fencing the Ground Ceil and Floor in C Compound Interest Program in C Displaying Array in C Distance Vector Routing Protocol Program in c Dos.h Header File in C Language DSA Program in C Explain the two-way selection in C Fee Management System in C File Operations in C Malloc function in C Multiplication Table in C Simple Programs in C Language tolower() Function in C Type Conversion in the C Why does sizeof(x++) not Increment x in C Advantages of Dynamic Memory Allocation in C Armstrong Number in C Assignment Operator Program in C Banker’s Algorithm in C Binary Search in C with Best and Worst Time Complexity Caesar Cipher Program in C Call by Value and Call by Reference in C Conditional Operator in C CRC Program in C Deadlock Detection Program in C Decimal to Binary in C Difference between If Else and Nested If Else in C Difference between Pre-increment and Post-increment in C Difference between Scope and Lifetime in C Evaluation of Arithmetic Expression in C Explain the Increment and Decrement Operators in C Fseek Function in C Functions in C How to Find Square Free Numbers in C Length of an Array Function in C OpenGL in C Projects on C language in 2023 Purpose of a Function Prototype in C Stdio.h in C Two-Dimensional array in C What is String Comparison in C C Compilers for Windows Functions and Recursion in C How to Declare Boolean in C How to Declare Character in C How to Round up a number in C How to use strlen() in C Pointer Declaration in C Algorithm for String Palindrome in C C Program to find ASCII value of a character Constant Pointer in C How to find string length in C using strlen() function Implicit and Explicit in C Indirect Recursion in C Input and Output functions in C isupper() in C Jump Statement in C Lifetime of a Variable in C Linker Error in C Language Numeric Constant in C Size of Pointer in C Square Root in C Language Static and Dynamic Memory allocation String Declaration in C Strong Number in C Symmetric Matrix in C Types of C Tokens What is a Size of Pointer in C What is Increment and Decrement Operator in C 1 2 3 4 Series Program in C Advantages and Disadvantages of C Language C Program for Polynomial Addition C Program to Count the Number of Vowels in a String C Programming Errors and Solutions Compilation Errors in C Complex C Programs Difference between Argument and Parameter in C Difference between char s[] and char *s in C Evaluation of Postfix Expression Using Stack in C Find Leftmost and Rightmost Set Bit of a Number fprintf and fscanf in C Introduction to Dynamic Array in C Print Address in C Realloc function in C Ternary Operators in C Types of Tokens in C with Examples Difference between Static and Dynamic Memory Allocation in C Addition Program in C Array Definition in C Array of Pointers in C Arrow Operator in C Average of Two Numbers in C Binary to Decimal in C Binary to Octal in C BREAK STATEMENT in C C Programming Operators Questions C Programs Asked in Interview Calculator Program in C C Program to Read and Print an Employee's Detail Using Structure Bubble Sort Algorithm in C C Program to Find Area and Perimeter of Circle C Program to Check Whether a Given Number is Even or Odd C in Roman Numerals C Program to Make a Simple Calculator Using Switch Case Insertion Sort Program in C How to take input in string in C GCC Conflicting Types in C Function Definition in C Format Specifier for Hexadecimal in C Flowchart in C Float in C Fizzbuzz Implementation in C Conditional Statement in C Conio.h functions list in C Constants in C Dynamic Array in C Decision Making Statements in C Continue Statement in C Creation of Thread in C DFS Algorithm in C Difference between parameter and arguments in C Dijkstra's Algorithm in C Leap Year Program in C Jump Statements in C Modulus Operator in C Memory Allocation in C Simple Interest Program in C Reverse Array in C Recursive Function in C Queue in C Printing Pascal’s Triangle in C Preprocessor Directives in C Perror() in C Perfect Number in C Programming Language Parameter Passing Techniques in C Pascal Triangle in C Patterm Program in C Affine cipher in C Dereferencing pointer in C Internal static variable vs External static variables in C Difference between exit(0) and exit(1) in C Booth's Algorithm in C Condition Control Statements in C Double Specifier in C Dynamic variables in C How to print alphabets in C How to print char array in c Order of Evaluation in C Order of Operations in C Semantic Error in C Size of String Variable in C SJF PREEMPTIVE SCHEDULING PROGRAM C: Tree in C Arithmetic Progression Program in C Array, Declaring Arrays, and Array Initialization ARRAYS IN C Assert() Function in c Atoi in C Banker’s Algorithm in C Bar3d() function in C Graphics Beep function in c Bigint (BIG INTEGERS) in C with Example Builtin functions of GCC compiler Fibonacci series in C Priority Queue in C 2D ARRAY IN C 7 Best IDEs for C/C++ Developers in 2022 Addition of Two Numbers in C Advantages and Disadvantages of C Language Advantages of Function in C Algorithm for String Palindrome in C

How to Find Square Free Numbers in C?

Introduction

Square-free number program is a typical interview and academic question in C Programming. In this section, we will define square-free integers, construct algorithms and C program to print the nth square-free number and find all the Square-Free Numbers in a Range. Further, we will also analyze their complexities.

What is Square Free Number?

A square-free number is an integer that is not divisible by the square of any prime number. In other words, a square-free number has no repeated prime factors.

For example, consider the prime numbers 2, 3, 5, and 7. These numbers are square-free because they are prime numbers and have no factors other than one and themselves. The number 15 is also square-free because it can be written as the product of prime numbers 3 and 5, and neither 3 nor 5 is a perfect square. The number 12, on the other hand, is not square-free because it can be written as the product of the prime numbers 2 and 3, and 2 is repeated.

Other examples include:

4: This number can be written as 2*2, and 2 is repeated prime. Therefore, 4 is not square-free.

9: This number can be written as 3*3, and 3 is a repeated prime. Therefore, 9 is not square-free.

27: This number can be written as 3*3*3, and 3 is repeated prime. Therefore, 27 is not square-free.

28: This number can be written as 2*2*7, and neither 2 nor 7 is a perfect square. Therefore, 28 is square-free.

30: This number can be written as 2*3*5, and neither 2, 3, nor 5 is a perfect square. Therefore, 30 is square-free.

Square-free numbers have unique factorization, which means that every positive integer can write as a product of prime numbers in only one way. This property is not shared by all integers, as some numbers (12) can be written as a product of prime numbers in multiple ways. For example, 12 can be written as 2*2*3 or 3*4; these two factorizations are not equivalent.

Square-free numbers are essential in mathematics because they have unique factorization, which makes them easier to work with in many contexts. They also have some unique properties that make them useful in number theory and other areas of mathematics. For example, the number of divisors of a square-free number is equal to the product of the exponents of its prime factors plus 1. 

For example, the number 15 has 2 prime factors (3 and 5), 3 has 2 (30 and 31) number of exponent, 5 also has 2 (50 and 51) number of exponent, and the number of divisors of 15 is (2 )( 2 ) = 4.

Similarly, the number 28 has 2 prime factors (2 and 7), 2 has 2 (20 and 21 ) exponent, and 7 also has 2 (70 and 71 ) exponent, and the number of divisors of 28 is ( 2 )( 2 ) = 4.

The number of divisors of a non-square-free number is not necessarily equal to the product of the exponents of its prime factors plus 1. For example, the number 12 has 2 prime factors (2 and 3), and the number of divisors of 12 is (2+1)(1+1) = 6.

Square-free numbers are less common than non-square-free numbers, but they have important applications in number theory and other areas of mathematics. For example, the Möbius function, which is a function that assigns a value of -1, 0, or 1 to each positive integer based on its prime factorization, is closely related to square-free numbers. The Möbius function is often used in combinatorial and probabilistic calculations, and it can be computed efficiently using square-free numbers.

C Program to Find Square Free Number

Algorithm

Here is an algorithm for finding square-free numbers in C step-by-step-

  1. Write a function that takes an integer as input and returns a Boolean indicating whether the number is square-free.
  2. Initialize a loop variable i to 2, which is the smallest prime number. The loop will iterate through all the integers from 2 to the square root of the input number or input number.
  3. Inside the loop, check if the input number n is divisible by i. If it is, check if n is divisible by the square of i. If it is, return false (not square-free).
  4. If the loop completes without returning false, return true (square-free).

Complexity

1. Time Complexity:

The time complexity of an algorithm refers to how long it takes for the algorithm to run as the size of the input increases. In the case of finding square-free numbers in C, the time complexity depends on the algorithm used to determine whether a given number is square-free or not.

One algorithm for finding square-free numbers involves checking whether a number is divisible by the squares of all the prime numbers less than or equal to the square root of the number. This algorithm has a time complexity of O(sqrt(n)), where n is the size of the input (the number being tested for square-freeness). It means that the time taken by the algorithm increases as the square root of the input size.

Another algorithm for finding square-free numbers involves checking whether a number is divisible by the squares of all the prime numbers less than or equal to the number. This algorithm has a time complexity of O(n), meaning that the time taken by the algorithm increases linearly with the input size.

Overall, the time complexity of finding square-free numbers in C depends on the specific algorithm used, but it is generally on the order of O(sqrt(n)) or O(n).

2. Space Complexity:

The space complexity of the algorithm refers to how much memory the algorithm needs to run as the input size increases. In the case of finding square-free numbers in C, the space complexity is typically determined by the amount of memory needed to store the input and any intermediate variables or data structures used by the algorithm.

For example, the space complexity of an algorithm that finds square-free numbers by checking whether a number is divisible by the squares of all the prime numbers less than or equal to the square root of the number is O(1) since it only requires a constant amount of memory to store the input and the loop variable.

On the other hand, an algorithm that finds square-free numbers by checking whether a number is divisible by the squares of all the prime numbers less than or equal to the number may have a higher space complexity, depending on how the list of prime numbers is stored. If the list of prime numbers is stored in an array, the space complexity would be O(n) since the array size would increase linearly with the input size.

Overall, the space complexity of finding square-free numbers in C depends on the specific algorithm used and how it handles the input and intermediate data.

Check if the Number is a Square-Free Number or Not  in C:

#include <math.h>
#include <stdbool.h>


bool is_square_free(int n) {
  // Iterate through all integers from 2 to the square root of n
  for (int i = 2; i <= sqrt(n); i++) {
    // Check if n is divisible by i
    if (n % i == 0) {
      // Check if n is divisible by the square of i
      if (n % (i * i) == 0) {
        return false;  // Not square-free
      }
    }
  }
  return true;  // Square-free
}


int main() {
  // Test the function with some sample inputs
  printf("%d\n", is_square_free(2));  // Output: 1 (square-free)
  printf("%d\n", is_square_free(4));  // Output: 0 (not square-free)
  printf("%d\n", is_square_free(9));  // Output: 0 (not square-free)
  printf("%d\n", is_square_free(27));  // Output: 0 (not square-free)
  printf("%d\n", is_square_free(28));  // Output: 1 (square-free)
  return 0;
}

Output:-

1
0
0
0
0

Find nth Square Free Number in C

Algorithm:

Here is an algorithm for finding the nth square-free number in C in step-by-step detail with an example:

  1. Write a function that takes an integer as input and returns a boolean indicating whether the number is square-free.
  2.  Initialize a loop variable i to 2, which is the smallest prime number. The loop will iterate through all the integers from 2 to the square root of the input number.
  3. Inside the loop, check if the input number n is divisible by i. If it is, check if n is divisible by the square of i. If it is, return false (not square-free).
  4. If the loop completes without returning false, return true (square-free).
  5. Write a main function that takes an integer n as input and returns the nth square-free number.
  6. Initialize a counter variable i to 1 and a variable result to 2, which is the first square-free number.
  7. While i is less than n, increment i and add result by 1. Check if the result is square-free using the is_square_free function. If it is not, continue incrementing the result until it is square-free.
  8. Return the result as the nth square-free number.

Complexity:

1. Time Complexity:-

In the case of finding the nth square-free number, the size of the input is n, the number of the square-free number we want to find.

One way to find the nth square-free number is to generate all of the positive integers in order and check each one to see if it is square-free. It can be done using a simple loop, starting at 1 and incrementing by 1 until we reach the nth square-free number.

The time complexity of this algorithm is O(n) since the number of operations we need to perform is directly proportional to n. The algorithm's running time will grow linearly with the input size.

It is a relatively straightforward approach, but there may be more efficient ways to find the nth square-free number. Other algorithms may have a lower time complexity, such as using a sieve to pre-compute the square-free numbers and then looking up the nth square-free number in the sieve. The time complexity of this approach will depend on the specific sieve algorithm used.

2. Space Complexity:

The space complexity of this algorithm is a measure of how much memory requires to run as a function of input size. In the case of finding the nth square-free number, the size of the input is n, the number of the square-free number we want to find.

One way to find the nth square-free number is to generate all of the positive integers in order and check each one to see if it is square-free. It can be done using a simple loop, starting at 1 and incrementing by 1 until we reach the nth square-free number.

The space complexity of this algorithm is O(1) since we only need to store a few variables (e.g., a counter and the current number we are checking) in memory, regardless of the input size. It means that the amount of memory required by the algorithm is constant and does not depend on the input size.

It is a relatively simple and straightforward approach, but there may be more efficient ways to find the nth square-free number. Other algorithms may have a lower space complexity, such as using a sieve to pre-compute the square-free numbers and then looking up the nth square-free number in the sieve. The space complexity of this approach will depend on the specific sieve algorithm used.

Print nth Square Free Numbers in C:-

#include <math.h>
#include <stdio.h>
#include <stdbool.h>


int main() 
{
  int n;
  printf("Enter a number: ");
  scanf("%d", &n);


  // Find the nth square-free number
  int i = 1;
  int result = 2;  // The first square-free number is 2
  while (i < n) {
    i++;
    result++;
    bool square_free = true;
    // Check if result is divisible by any number from 2 to sqrt(result)
   
 for (int j = 2; j <= sqrt(result); j++) {
      if (result % j == 0) {
        square_free = false;
  break;
        }
      }
    
// If result is not square-free, continue incrementing until it is
    while (!square_free) {
      result++;
      square_free = true;
      for (int j = 2; j <= sqrt(result); j++) {
        if (result % j == 0) {
          square_free = false;
          break;
        }
      }
    }
  }
  printf("The %dth square-free number is %d\n", n, result);
  return 0;
}

Output:-

Enter a number: 100
The 100th square-free number is 541

Finding All the Square-Free Numbers Within Given Range

Algorithm:

To write an algorithm for finding all the square-free numbers within a given range in detail, you can follow these steps:

  1. Initialize a loop variable i to the lower bound of the range. The loop will iterate through all the integers from a lower bound to an upper bound.
  2. Inside the loop, check if i is square-free by iterating through all the integers from 2 to the square root of i, checking if i is divisible by each of them. If i is divisible by any of them, it means that i is not square-free, and you can continue to the next iteration of the loop.
  3. If i is not divisible by any of the integers from 2 to the square root of i, it means that i is square-free, and you can print it as a result.
  4. Repeat steps 2 and 3 until the loop reaches the range's upper bound.

Complexity

1. Time Complexity:

In this case, the input is the range of integers being checked for square-freeness.

The algorithm for finding all the square-free numbers within a given range has a time complexity of O(n*sqrt(n)), where n is the size of the input (the range of integers being checked). It is because the algorithm involves a loop that iterates through all the integers in the range (n iterations), and for each integer, it checks whether it is divisible by any of the integers from 2 to the square root of the integer (sqrt(n) iterations).

In other words, the time taken by the algorithm increases as the product of the size of the input (n) and the square root of the size of the input (sqrt(n)). The algorithm's time complexity is slower than linear but not quite quadratic.

2. Space Complexity:

The space complexity of this algorithm refers to how much memory the algorithm uses as a function of input size. In this case, the input is the range of integers being checked for square-freeness.

The algorithm for finding all square-free numbers within a given range has a space complexity of O(1), and the algorithm's memory usage is constant no matter the input size.

The algorithm does not store data or create new variables beyond a few simple variables for looping and storing intermediate results.

In this case, the space complexity of the algorithm is independent of the size of the input and remains constant regardless of the size of the range checked. It makes the algorithm very efficient in terms of memory usage.

C Program of Finding All the Square-Free Numbers Within a Given Range:-

#include <math.h>
#include <stdio.h>
#include <stdbool.h>


int main() 
{
  int lower, upper;
  printf("Enter the lower bound of the range: ");
  scanf("%d", &lower);
  printf("Enter the upper bound of the range: ");
  scanf("%d", &upper);


  // Iterate through all integers from lower to upper
  for (int i = lower; i <= upper; i++) {
    // Initialize a flag to indicate whether i is square-free or not
    bool square_free = true;
    // Iterate through all integers from 2 to sqrt(i)
    for (int j = 2; j <= sqrt(i); j++) {
      // If i is divisible by j, it is not square-free
      if (i % j == 0) {
        square_free = false;
        // Break out of the inner loop, as there is no need to check further
        break;
      }
    }
    // If i is square-free, print it
    if (square_free) {
      printf("%d\n", i);
    }


} 
return 0;
 }


Output:-

Enter the lower bound of the range: 1
Enter the upper bound of the range: 100


1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 

Conclusion

In conclusion, finding square-free numbers in C involves checking whether a number is divisible by the squares of all the prime numbers less than or equal to the square root of the number. It can do it by iterating through all the integers from 2 to the square root of the number, checking if the number is divisible by each of them. If the number is divisible by any of them, it is not square-free. Moreover, to find the square-free numbers within the given range; you can use the loop to iterate through all the integers in the range and apply this check to each of them. To find the nth square-free number, you can use a similar algorithm but stop once you have found the desired number of square-free numbers. The time complexity of these algorithms depends on the size of the input range and the number of prime numbers within it.