  Sure, here's a Python code that should perform the steps you described:
```
def square_root_algorithm(n):
    # Initialize array x with all positions set to True
    x = [True] * n

    # Change the values of the first two positions to False
    x[0] = False
    x[1] = False

    # Iterate from 2 to the nearest integer to the square root of n
    for j in range(2, int(n ** 0.5) + 1):
        # If the array x at position j takes the value True, update the values of x to False
        # at positions j*j through n by jumping from j to j
        if x[j]:
            for i in range(j * j, n, j):
                x[i] = False

    # Return the indices of the array x that take the value True
    return [i for i, j in enumerate(x) if j]
```
Here's a breakdown of how the code works:

1. The function `square_root_algorithm(n)` takes a natural number `n` as input.
2. The array `x` is initialized with all positions set to `True`. This is done using a list comprehension, which creates a list of `n` `True` values.
3. The first two positions of the array `x` are set to `False`. This is done using indexing, where `x[0]` and `x[1]` are assigned the value `False`.
4. The function then iterates from 2 to the nearest integer to the square root of `n` using a `for` loop. This is done using the `range` function, which creates a sequence of numbers from 2 to the specified number.
5. Inside the loop, the function checks whether the value of `x` at position `j` is `True`. If it is, then it updates the values of `x` to `False` at positions `j*j` through `n` by jumping from `j` to `j`. This is done using another `for` loop, which iterates over the range `j*j` to `n` in steps of `j`. Inside this loop, the value of `x` at the current index `i` is set to `False`.
6. Finally, the function returns the indices of the array `x` that take the value `True`. This is done using a list comprehension, which creates a list of indices `i` such that `x[i]` is `True`. The `enumerate` function is used to get the indices of the elements in the array.

Here's an example of how the function works for the input `n = 16`:

1. The array `x` is initialized with all positions set to `True`: `[True, True, True, True, True, True, True, True, True, True, True, True, True, True]`.
2. The first two positions of the array `x` are set to `False`: `[False, False, True, True, True, True, True, True, True, True, True, True, True, True]`.
3. The function iterates from 2 to the nearest integer to the square root of `n` (which is 4 in this case).
4. For `j = 2`, the value of `x` at position `j` is `True`, so the values of `x` at positions `j*j` through `n` are set to `False`: `[False, False, True, False, True, True, True, True, True, True, True, True, True, True]`.
5. For `j = 3`, the value of `x` at position `j` is `True`, so the values of `x` at positions `j*j` through `n` are set to `False`: `[False, False, True, False, True, False, True, True, True, True, True, True, True, True]`.
6. For `j = 4`, the value of `x` at position `j` is `True`, so the values of `x` at positions `j*j` through `n` are set to `False`: `[False, False, True, False, True, False, True, False, True, True, True, True, True, True]`.
7. The function returns the indices of the array `x` that take the value `True`: `[2, 3, 5, 7, 11, 13]`.

I hope this helps! Let me know if you have any questions or need further clarification.