Maximize XOR

Let $2^k\le p<q\le 2^{k+1}-1$

We are looking for pairs (p,q) such as
$\bullet p\otimes q$ is maximal // $\otimes$ is the XOR operator.
$\bullet p,q$ are both either a prime or a square number

How many such pairs are there for k = 26 ?

Answer format: count,first p,first q

Example: 3,1307,1764 for k = 10

[My timing: 42 sec]