TheAlgorithms-C-Plus-Plus/math/lcm_sum.cpp

/**
 * @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;
}