NOTES.md

Brute Force Approach

1. Function Definition: The code defines a Python function named sumOfAllDivisorsBruteForce which calculates the sum of all divisors of a given integer n. The function takes an integer n as its argument and returns an integer as the result.

2. Divisor List Initialization: An empty list named divisor_list is initialized. This list will be used to store the divisors of the numbers.

3. Outer Loop (Numbers 1 to n):

   • A for loop iterates through numbers in the range from 1 to n, represented by the variable num.

4. Inner Loop (Possible Divisors):

   • For each num, an inner for loop (for j in range(1, int(num**0.5) + 1)) is used to iterate through possible divisors.
   • Within this inner loop, it checks if j is a divisor of num by calculating num % j. If the result is zero, it means j is a divisor of num.

5. Adding Divisors to List:

   • If j is a divisor of num, it is added to the divisor_list.
   • Additionally, to ensure all divisors are included correctly, it checks if j is not equal to num/j before adding it to the list. This accounts for both divisors on each side of the square root (if applicable).

6. Sum Calculation: After both loops have run for all numbers from 1 to n, the code calculates the sum of all the divisors in the divisor_list using the sum() function.

7. Returning the Result: The sum of all divisors is returned as the result of the function.

Time Complexity Analysis

1. Outer Loop:

   • The outer for loop iterates from 1 to n. This loop runs exactly n times.

2. Inner Loop:

   • For each value of num in the outer loop, the inner for loop iterates from 1 to the square root of num, which is approximately sqrt(n) iterations for each num.

3. Divisor Checking:

   • Within the inner loop, the code checks whether num is divisible by j (i.e., num % j == 0).

4. Appending to List:

   • For each valid divisor found, the code appends it to the divisor_list. In the worst case, both divisors (if they exist) will be appended, so this operation is constant time.

5. Sum Calculation:

   • After both loops have run for all numbers from 1 to n, the code calculates the sum of all the divisors using the sum() function. This is a linear operation, as it sums all elements in the divisor_list.

As a result, the overall time complexity of the code is dominated by the nested loops:

• The outer loop runs n times.
• The inner loop, for each num, runs approximately sqrt(num) times.

Therefore, the worst-case time complexity of this code is approximately $O(n * sqrt(n))$, where n is the input integer. This is because it iterates through numbers from 1 to n and, for each number, iterates up to the square root of that number while performing constant-time operations for each valid divisor.

The time complexity of O(n * sqrt(n)) indicates that the code's runtime will increase roughly in proportion to the square root of the input integer n. As 'n' grows larger, the execution time will increase significantly.

Space Complexity Analysis

1. divisor_list (List):

   • An empty list named divisor_list is initialized to store the divisors of the numbers. The size of this list grows as divisors are found and added to it.
   • In the worst case, when 'n' is prime, each number from 1 to 'n' may have two divisors (itself and 1), resulting in approximately 2 * 'n' elements being added to the list.
   • Therefore, the space complexity attributed to divisor_list is O(n).

2. Variables (int):

   • The code uses a few integer variables such as num, j, and the loop control variables. These variables occupy constant space, and their space usage does not depend on the input 'n'.
   • Therefore, the space complexity due to integer variables is O(1).

3. Total Space Complexity:

   • The dominant factor contributing to space complexity is the divisor_list, which is O(n).
   • The other variables use a constant amount of space, O(1).

As a result, the overall space complexity of the code is $O(n)$ due to the space used by the divisor_list. This indicates that the amount of memory required by the code grows linearly with the size of the input integer n. When n is large, the space used by the list can become a significant factor in terms of memory consumption.