A note on the products (μ1+1)(μ2+1)⋯(nμ+1)

Let Ωμ(n)=(μ1+1)(μ2+1)⋯(nμ+1) where μ⩾2 is an integer. We prove that Ω3(n) is never squarefull, and in particular never a square, using arguments similar to those in J. Cilleruelo (2008) [2], , where it is proven that Ω2(n) is not a square for n≠3. In T. Amdeberhan et al. (2008) [1], among many other results, it is claimed that Ωμ(n) is not a square if μ is an odd prime and n>12. However, we have found a gap in the proof of this statement, which we illustrate by giving counterexamples.