When $(n,k) \in \mathbb P^2 $, the following coprime relations appear to hold:
$$\gcd\Bigl(n^k,{\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor}^{k}\Bigr)=1$$
$$\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor,{\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor}^{k}\Bigr)=1$$
So the first part of my question is a request for a counter example for these, which are proving to be more and more elusive. As far I know thus far, they can also be extended to $(n,k)$ for which $k$ is any natural number.
however the later term in the greatest common divisor expression above ${\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor}^{k}$, is not always coprime to $\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor$ and the strictly prime domain of $(n,k)$ is paritioned into two equivalence classes, the predicates for which are the coprimality of this pair of quantities and it's negation: $$\mathbb P^2=\mathbb P_1 \cup \mathbb P_2$$
$$\mathbb P_1={\Biggl\{(n,k):\gcd\Bigl(\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor,{\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor}^{k}\Bigr)=1}\Biggr\}$$
$$\mathbb P_2={\Biggl\{(n,k):\gcd\Bigl(\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor,{\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor}^{k}\Bigr) \gt 1}\Biggr\}$$
A sample of my numerical observations thus far are:
$$\mathbb P_1={\{(2, 2), (2, 3), (2, 5), (2, 7), (3, 2), (3, 3), (3, 5), (3, 7), (5, 5), (5, 7), (7, 2), (11, 2), (11, 5), (11, 7)}\}$$
$$\mathbb P_2={\{(5, 2), (5, 3), (7, 3), (7, 5), (7, 7), (11, 3), (13, 2), (13, 5), (17, 2), (19, 2), (19, 7), (29, 5)}\}$$
likewise,$\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor$ and $n^k$ are not always coprime, and the strictly prime domain of $(n,k)$ is paritioned into two equivalence classes, the predicates for which are the coprimality of this pair of quantities and it's negation:
$$\mathbb P^2=\mathbb P_1 \cup \mathbb P_2$$
$$\mathbb P_1={\{(n,k):\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor\Bigr)=1}\}$$
$$\mathbb P_2={\{(n,k):\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n^k}{n^k} \Bigr\rfloor\Bigr) \gt 1}\}$$
Numerical observations thus far are:
$$\mathbb P_1={\{(2, 3), (2, 5), (3, 2), (3, 3), (5, 2), (5, 5), (7, 2), (11, 2), (11, 5), (13, 2), (13, 3), (17, 2), (17, 3)}\}$$
$$\mathbb P_2={\{(2, 2), (3, 5), (5, 3), (7, 3), (7, 5), (11, 3), (13, 5), (23, 3)}\}$$
So the second part of my question, is how, and using what theorem, can I express the above as a congruence relation?
I have tried looking for the sequence of values of $n$ for various fixed values of $k$ on OEIS, but nothing has shown up. I'm not sure if my problem is important enough to put these sequences on there or not, it asks me to do so if I think they are of interest, but that's pretty subjective, I mean if it really is like face book and sequence popularity matters, anyway end of question.