To determine which bulbs remain ON at the end of the given exercise, we observe the following:

1. Each bulb $k$ is toggled on each step corresponding to its divisors. For instance:

   • Bulb 12 is toggled for steps 1, 2, 3, 4, 6, and 12 (its divisors).

2. A bulb ends up ON only if it is toggled an odd number of times. A number has an odd number of divisors only if it is a perfect square. For example:

   • $16$ has divisors $1, 2, 4, 8, 16$, which is odd.
   • $12$ has divisors $1, 2, 3, 4, 6, 12$, which is even.

3. Hence, the problem reduces to finding all perfect squares from $1$ to $500$.

Algorithm:

1. Initialize a list of size $500$ with all bulbs set to "OFF."
2. Loop through each number from 1 to 500:
   • Toggle the bulbs corresponding to multiples of the current number.
3. Return the bulbs that remain ON (those corresponding to perfect squares).

Optimized Insight:

The bulbs remaining ON are only those at positions corresponding to perfect squares $n^2 \leq 500$. Compute $n$ such that $n^2 \leq 500$.

Steps to Find Bulbs:

1. List all integers $n$ such that $n^2 \leq 500$.
2. Compute the squares: $n = 1, 2, 3, \ldots, 22$ (since $22^2 = 484$).
3. Return $1^2, 2^2, \ldots, 22^2$.

Python Code:

pythonCopy code
def glowing_bulbs():
    perfect_squares = []
    n = 1
    while n**2 <= 500:
        perfect_squares.append(n**2)
        n += 1
    return perfect_squares

print(glowing_bulbs())

Output:

The bulbs that glow are: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484$.

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Would you like details about the implementation or mathematical reasoning? Here are related questions to deepen your understanding:

1. Why do only perfect squares have an odd number of divisors?
2. How can you generalize this algorithm for a different number of bulbs or steps?
3. What is the time complexity of the naive algorithm versus the optimized one?
4. How does the structure of divisors relate to toggling patterns?
5. Can this algorithm be extended for prime-based toggle steps?

Tip: To identify perfect squares in a range, consider the sequence of integers squared from 1 upwards.