Can anyone help me with a technical mathematical divisibility issue?

This is a technical issue that came up in my research, and I could probably program something, but I wonder if anyone has any insight. Given a fixed natural number d (which I’m generally assuming is squarefree) I’m looking for examples of positive integers r, k, such that

1. r is prime
2. r divides d*k^2–1, say rs=d*k^2–1, and (r,s)=1
3. k<r.

It seems like there should be plenty of examples of this situation, but I’m having trouble drumming them up, and even more trouble getting a method to generate examples. I’d appreciate any nontrivial examples.