Primefree shifted Lucas sequences of the second kind

We say a sequence S=(sn)n≥0 is primefree if |sn| is not prime for all n≥0 and, to rule out trivial situations, we require that no single prime divides all terms of S. Recently, the second author showed that there exist infinitely many integers k such that both of the shifted sequences U±k are simultaneously primefree, where U is a particular Lucas sequence of the first kind. In this article, we prove an analogous result for the Lucas sequences Va=(vn)n≥0 of the second kind, defined byv0=2,v1=a,andvn=avn−1+vn−2,for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences Va±k are simultaneously primefree. This result provides additional evidence to support a conjecture of Ismailescu and Shim. Moreover, we show that there are infinitely many values of k such that every term of both of the shifted sequences Va±k has at least two distinct prime factors.