A note on the products (1(mu)+1)(2(mu)+1) ... (n(mu)+1)

Let Omega(mu)(n) = (1(mu) + 1)(2(mu) + 1) ... (n(mu) + 1) where mu >= 2 is an integer. We prove that Omega(3)(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 Omega(2)(n) is not a square for n not equal 3. In T. Amdeberhan et al (2008) [1], among many other results, it is claimed that Omega(mu)(n) is not a square if mu 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. (C) 2009 Elsevier Inc. All rights reserved.