Merge pull request #1344 from chestamittal/master

feat: add lcm_sum.cpp

DIRECTORY.md

@@ -142,6 +142,7 @@
   * [Gcd Recursive Euclidean](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/gcd_recursive_euclidean.cpp)
   * [Large Factorial](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/large_factorial.cpp)
   * [Large Number](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/large_number.h)
+  * [Lcm Sum](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/lcm_sum.cpp)
   * [Least Common Multiple](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/least_common_multiple.cpp)
   * [Miller Rabin](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/miller_rabin.cpp)
   * [Modular Inverse Fermat Little Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_fermat_little_theorem.cpp)

math/lcm_sum.cpp

@@ -0,0 +1,99 @@
+/**
+ * @file
+ * @brief An algorithm to calculate the sum of LCM: \f$\mathrm{LCM}(1,n) + \mathrm{LCM}(2,n) + \ldots + \mathrm{LCM}(n,n)\f$
+ * @details An algorithm to calculate the sum of LCM: \f$\mathrm{LCM}(1,n) + \mathrm{LCM}(2,n) + \ldots + \mathrm{LCM}(n,n)\f$ where \f$\mathrm{LCM}(i,n)\f$
+ * denotes the Least Common Multiple of the integers i and n. For n greater than or equal to 1.
+ * The value of the sum is calculated by formula:
+ * \f[
+ * \sum\mathrm{LCM}(i, n) = \frac{1}{2} \left[\left(\sum (d * \mathrm{ETF}(d)) + 1\right) * n\right]
+ * \f]
+ * where \mathrm{ETF}(i) represents Euler totient function of i.
+ * @author [Chesta Mittal](https://github.com/chestamittal)
+ */
+
+#include <iostream> /// for std::cin and std::cout
+#include <cassert>  /// for assert
+#include <vector>   /// for std::vector
+
+/**
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+    /**
+     * Function to compute sum of euler totients in sumOfEulerTotient vector
+     * @param num input number
+     * @returns int Sum of LCMs, i.e. ∑LCM(i, num) from i = 1 to num
+     */
+    uint64_t lcmSum(const uint16_t& num) {
+   
+        uint64_t i=0, j=0;
+        std::vector <uint64_t> eulerTotient(num+1);
+        std::vector <uint64_t> sumOfEulerTotient(num+1);
+   
+        // storing initial values in eulerTotient vector
+        for(i=1; i<=num; i++) {
+            eulerTotient[i] = i;
+        }
+
+        // applying totient sieve  
+        for(i=2; i<=num; i++) {
+            if(eulerTotient[i] == i) {
+                for(j=i; j<=num; j+=i) {
+                    eulerTotient[j] = eulerTotient[j]/i;
+                    eulerTotient[j] = eulerTotient[j]*(i-1);
+                }
+            }
+        }
+
+        // computing sum of euler totients
+        for(i=1; i<=num; i++) {
+            for(j=i; j <=num; j+=i) {
+                sumOfEulerTotient[j] += eulerTotient[i]*i;
+            }
+        }
+    
+        return ((sumOfEulerTotient[num] + 1 ) * num) / 2;
+    }
+}  // namespace math
+
+/**
+ * Function for testing lcmSum function.
+ * test cases and assert statement.
+ * @returns `void`
+*/
+static void test() {
+    uint64_t n = 2;
+    uint64_t test_1 = math::lcmSum(n);
+    assert(test_1 == 4);
+    std::cout << "Passed Test 1!" << std::endl;
+
+    n = 5;
+    uint64_t test_2 = math::lcmSum(n);
+    assert(test_2 == 55);
+    std::cout << "Passed Test 2!" << std::endl;
+
+    n = 10;
+    uint64_t test_3 = math::lcmSum(n);
+    assert(test_3 == 320);
+    std::cout << "Passed Test 3!" << std::endl;
+
+    n = 11;
+    uint64_t test_4 = math::lcmSum(n);
+    assert(test_4 == 616);
+    std::cout << "Passed Test 4!" << std::endl;
+
+    n = 15;
+    uint64_t test_5 = math::lcmSum(n);
+    assert(test_5 == 1110);
+    std::cout << "Passed Test 5!" << std::endl;
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test();  // execute the tests
+    return 0;
+}