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


Guerel E., KİŞİSEL A. U. Ö.

JOURNAL OF NUMBER THEORY, cilt.130, ss.187-191, 2010 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 130 Konu: 1
  • Basım Tarihi: 2010
  • Doi Numarası: 10.1016/j.jnt.2009.07.014
  • Dergi Adı: JOURNAL OF NUMBER THEORY
  • Sayfa Sayıları: ss.187-191

Özet

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.