> Please tell me how to vectorize this Python code. Go on, I'm waiting.

You can't entirely get rid of the loops, but at least the inner one can be vectorized, I think. That would reduce the "pythonic runtime" from squared to linear.

Maybe something like this?

  result = np.full((limit+1,), True, dtype=bool)
  result[0] = False
  indices = np.arange(len(result))
  for i in range(2, len(result)):
    result[indices[i:] * i] = False

This is just out of my head, so may contain bugs. Also, the indices array may need to be clamped, not sure right now if numpy accepts out-of-bounds indices.