Summary: We say a sequence \(\mathcal S=(s_n)_{n\geq 0}\) is primefree if \(|s_n|\) is not prime for all \(n\geq 0\), and to rule out trivial situations, we require that no single prime divides all terms of \(\mathcal S\). In this article, we focus on the particular Lucas sequences of the first kind, \(\mathcal U_a=(u_n)_{n\geq 0}\), defined by \[ u_0=0,\, u_1=1, \text{ and } u_n=au_{n-1}+u_{n-2} \text{ for } n\geq 2, \] 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 \(\mathcal U_a\pm k\) are simultaneously primefree. This result extends previous work of the author for the single shifted sequence \(\mathcal U_a-k\) when \(a=1\) to all other values of \(a\), and establishes a weaker form of a conjecture of \textit{D. Ismailescu} and \textit{P. C. Shim} [Integers 14, Paper A65, 12 p. (2014; Zbl 1343.11020)]. Moreover, we show that there are infinitely many values of \(k\) such that every term of both of the shifted sequences \({\mathcal U}_a\pm k\) has at least two distinct prime factors.