Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P, Q) such that the rank of N in U(P, Q) is exactly (N - epsilon(N))/d, where a is the signature of U(P, Q). We prove here that all but a finite number of Lucas d-pseudoprimes are square free. We also prove that all but a finite number of Lucas d-pseudoprimes are Carmichael-Lucas numbers.