It is not going to handle a vast range of inputs, including for example, p(k) = 665, for the basic reason that my code is a direct response to your request for

It is not going to handle a vast range of inputs, including for example, p(k) = 665, for the basic reason that my code is a direct response to your request for an explanation and not proposed as a complete solution for the request of your original submission. Corrections requested, please. I would like to explore this paradigm to accurately address the problem in an exact way.