[fix/docs]: fit armstrong_number.cpp to guidelines (TheAlgorithms#2457)

* fix: fit armstrong_number.cpp to guidelines

* chore: delete unnecessary whitespace

* updating DIRECTORY.md

* fit: aliquot_sum to contribution guidelines

added a math formula

* chore: add print statement mentioning tests have passed

* Update math/armstrong_number.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/armstrong_number.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

---------

Co-authored-by: github-actions[bot] <github-actions@users.noreply.github.com>
Co-authored-by: David Leal <halfpacho@gmail.com>

Author: 3 people

math/aliquot_sum.cpp

@@ -3,10 +3,18 @@
  * @brief Program to return the [Aliquot
  * Sum](https://en.wikipedia.org/wiki/Aliquot_sum) of a number
  *
- * \details
- * The Aliquot sum s(n) of a non-negative integer n is the sum of all
+ * @details
+ * The Aliquot sum \f$s(n)\f$ of a non-negative integer n is the sum of all
  * proper divisors of n, that is, all the divisors of n, other than itself.
- * For example, the Aliquot sum of 18 = 1 + 2 + 3 + 6 + 9 = 21
+ *
+ * Formula:
+ *
+ * \f[
+ *      s(n) = \sum_{d|n, d\neq n}d.
+ * \f]
+ *
+ * For example;
+ *  \f$s(18) = 1 + 2 + 3 + 6 + 9 = 21 \f$
  *
  * @author [SpiderMath](https://github.com/SpiderMath)
  */
@@ -19,8 +27,9 @@
  * @namespace math
  */
 namespace math {
+
 /**
- * Function to return the aliquot sum of a number
+ * @brief to return the aliquot sum of a number
  * @param num The input number
  */
 uint64_t aliquot_sum(const uint64_t num) {
@@ -63,6 +72,5 @@ static void test() {
  */
 int main() {
     test();  // run the self-test implementations
-
     return 0;
 }

math/armstrong_number.cpp

@@ -1,20 +1,27 @@
 /**
- * @file
- * \brief Program to check if a number is an [Armstrong/Narcissistic
- * number](https://en.wikipedia.org/wiki/Narcissistic_number) in decimal system.
- *
- * \details
- * Armstrong number or [Narcissistic
- * number](https://en.wikipedia.org/wiki/Narcissistic_number) is a number that
- * is the sum of its own digits raised to the power of the number of digits.
- * @author iamnambiar
- */
-#include <cassert>
-#include <cmath>
-#include <iostream>
+* @file
+* @brief Program to check if a number is an [Armstrong/Narcissistic
+* number](https://en.wikipedia.org/wiki/Narcissistic_number) in decimal system.
+*
+* @details
+* Armstrong number or [Narcissistic
+* number](https://en.wikipedia.org/wiki/Narcissistic_number) is a number that
+* is the sum of its own digits raised to the power of the number of digits.
+*
+* let n be the narcissistic number,
+* \f[F_b(n) = \sum_{i=0}^{k-1}d_{i}^{k}\f] for
+* \f$ b > 1 F_b : \N \to \N \f$ where
+* \f$ k = \lfloor log_b n\rfloor is the number of digits in the number in base \f$b\f$, and
+* \f$ d_i = \frac{n mod b^{i+1} - n mod b^{i}}{b^{i}} \f$
+*
+* @author [Neeraj Cherkara](https://github.com/iamnambiar)
+*/
+#include <cassert> /// for assert
+#include <cmath> /// for std::pow
+#include <iostream> /// for IO operations
 
 /**
- * Function to calculate the total number of digits in the number.
+ * @brief Function to calculate the total number of digits in the number.
  * @param num Number
  * @return Total number of digits.
  */
@@ -28,16 +35,17 @@ int number_of_digits(int num) {
 }
 
 /**
- * Function to check whether the number is armstrong number or not.
- * @param num Number
+ * @brief Function to check whether the number is armstrong number or not.
+ * @param number to be checked
  * @return `true` if the number is armstrong.
  * @return `false` if the number is not armstrong.
  */
 bool is_armstrong(int number) {
-    // If the number is less than 0, then it is not a armstrong number.
+    // If the number is less than 0, then it is not an armstrong number.
     if (number < 0) {
         return false;
     }
+
     int sum = 0;
     int temp = number;
     // Finding the total number of digits in the number
@@ -46,17 +54,17 @@ bool is_armstrong(int number) {
         int rem = temp % 10;
         // Finding each digit raised to the power total digit and add it to the
         // total sum
-        sum = sum + std::pow(rem, total_digits);
+        sum += static_cast<int>(std::pow(rem, total_digits));
         temp = temp / 10;
     }
     return number == sum;
 }
 
 /**
- * Function for testing the is_armstrong() function
- * with all the test cases.
- */
-void test() {
+* @brief Self-test implementations
+* @returns void
+*/
+static void test() {
     // is_armstrong(370) returns true.
     assert(is_armstrong(370) == true);
     // is_armstrong(225) returns false.
@@ -69,12 +77,15 @@ void test() {
     assert(is_armstrong(0) == true);
     // is_armstrong(12) returns false.
     assert(is_armstrong(12) == false);
+
+    std::cout << "All tests have successfully passed!\n";
 }
 
 /**
- * Main Function
- */
+* @brief Main Function
+* @returns 0 on exit
+*/
 int main() {
-    test();
+    test(); // run self-test implementations
     return 0;
 }