chore: import upstream snapshot with attribution

math/lcm_sum.cpp

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+/**
+ * @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 <cassert>   /// for assert
+#include <cstdint>
+#include <iostream>  /// for std::cin and std::cout
+#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;
+}