Keith Numbers In C

A Keith number, also known as a Repfigit number, belongs to a category of numbers distinguished by a unique characteristic related to their individual digits. These numbers are part of a specific sequence where each number is derived by adding up its preceding digits. As a result, a number earns the title of a Keith or Repfigit number when it satisfies the following sequence criteria:

For each n-digit number, a sequence is formed using its digits. Each subsequent term is the sum of the preceding n terms. When a number generates itself within the sequence, it is known as a Keith number.

Algorithm:

✅ Start with n.
✅ Create an empty vector known as elements in which the sequence is to be stored.
✅ Count the number of digits and fill in each digit in the vector.
✅ Reverse the digits in the vector.
✅ Call next-element=0.
✅ The process proceeds with a loop until the exponential function falls below n.
✅ Add the last n digits to get the next element.
✅ Include the next element as part of the vector.
✅ Return true if the next element is equal to n; return false instead.

Steps

Begin by setting up an integer variable based on the provided instructions or input from the user.

Step 2: Generate an array to store the calculated values.

Step 3: Following this, determine the quantity of numerical digits present in the provided input number.

Step 4: The algorithm inside the loop is utilized to calculate consecutive numbers until surpassing the input number.

Finally, it is crucial to verify if the last computed number matches the initial input number. If the two numbers are indeed equal, we can confidently determine that the input number qualifies as a Keith Number.

Advantages of Keith Numbers:

Several advantages of Keith Numbers in C are as follows:

✅ Mathematical Interest: Keith's numbers are unusual and behave interestingly in the sequence, which makes them a good point of interest in recreational mathematics.
✅ Algorithm Design: These numbers provide test cases for the development and optimization of algorithms concerning digit extraction, sequence generation, and simple array manipulation.
✅ Educational Resonances: Keith's numbers provide an excellent standpoint to present introductory programming, recursion, iterations, and mathematical computation concepts
✅ Complex Algorithm Problem Solution: They appear in competitive programming, thereby requiring participants to merge problem-solving skills.
✅ Research-worthy: Keith's numbers help in the research of number theory.

Disadvantages of Keith Numbers:

Several advantages of Keith Numbers in C are as follows:

✅ Limited Usage: The vast majority of Keith's numbers are usually deemed theoretical, and applications are mainly limited to educational mathematics.
✅ High Computational Cost: As the numbers become large, generating the sequence becomes rather expensive in terms of computation, especially with respect to multi-digit Keith numbers.
✅ Very Scarce: Due to the dispersed nature of Keith numbers, locating a Keith number becomes very counter-productive.
✅ Sensibility of Algorithm: Computation of big numbers or exhaustive searching for all Keith numbers in the range will go into precious computing cycles, and it runs out of steam on a large scale.
✅ Very Niche: The study of Keith's numbers is niche enough that it does not attract a wider audience interested in the subject.

Example:

Let's consider an example to demonstrate the concept of a Keith number in the C programming language.

Example


//Program to check whether a given number is Keith number or not in C
#include <stdio.h> 
#include <stdlib.h>
 
//The function to count the length
int digCount(int number)
{
  int counter = 0;
  while (number > 0)
  {
    number = number / 10;
    counter++;
  }
  return counter;
}
int main()
{
  int n1 = 0, array1[10], t = 0, flagpVal = 0, i = 0, total = 0;
  printf("Please enter the number: ");
  scanf("%d", &n1);
  t = n1;
  for (i = digCount(t) - 1; i >= 0; i--)
  {
    array1[i] = n1 % 10;
    n1 /= 10;
  }
  while (flagpVal == 0)
  {
    for (i = 0; i < digCount(t); i++)
      total += array1[i];
    if (total == t)
    {
      printf("The given number is a Keith Number\n");
      flagpVal = 1;
    }
    if (total > t)
    {
      printf("The given number is NOT a Keith number\n");
      flagpVal = 1;
    }
    for (i = 0; i < digCount(t); i++)
    {
      if (i != digCount(t) - 1)
        array1[i] = array1[i + 1];
      else
        array1[i] = total;
    }
    total = 0;
  }
}

Output:

Output


Please enter the number: 45
The given number is NOT a Keith number

Explanation:

In this illustration, the software verifies whether a given number qualifies as a Keith number. Initially, it determines the count of digits by utilizing the digCount function, which calculates this count through iterative division of the number by 10. The script then compiles these digits into an array and establishes a sequence where each subsequent term is the sum of the preceding digits. Subsequently, the collected digits are arranged from right to left and stored in reverse within an array named array1. This array combines the number's digits with the dynamically formed sequence. The dynamic sequence is continuously updated by cycling through each element in the array and appending the new term at the conclusion. If the initial number is present within the generated sequence of terms, it indicates it is a Keith number; otherwise, it does not meet the criteria. This process persists until the collection surpasses or matches the user's specified input.

Conclusion:

In summary, Keith Numbers offer a unique opportunity for enhancing problem-solving abilities, testing algorithm design, and practicing programming skills. Despite being rare, they are intriguing from a mathematical perspective in programming as they involve manipulating digits, generating sequences, and fostering algorithmic reasoning. Keith numbers are infrequent occurrences, mainly serving as an academic challenge due to their computational complexity with larger numbers. Nevertheless, their distinct characteristics spark curiosity within recreational mathematics.

Keith Numbers are a unique set of numbers where their individual digits are used to create a specific sequence when organized in a particular way. The process involves checking if a number qualifies as a Keith number. This involves extracting the digits of the number and storing them in an array. The sequence is then produced by summing the prior n digits (where n represents the number of digits) repeatedly until the total matches or surpasses the original number. A number is identified as a Keith number if it emerges within this sequence.