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On strings of consecutive integers with a distinct number of prime factors


Authors: Jean-Marie De Koninck, John B. Friedlander and Florian Luca
Journal: Proc. Amer. Math. Soc. 137 (2009), 1585-1592
MSC (2000): Primary 11A25, 11N64
DOI: https://doi.org/10.1090/S0002-9939-08-09702-5
Published electronically: November 18, 2008
MathSciNet review: 2470816
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\omega (n)$ be the number of distinct prime factors of $n$. For any positive integer $k$ let $n=n_k$ be the smallest positive integer such that $\omega (n+1),\ldots ,\omega (n+k)$ are mutually distinct. In this paper, we give upper and lower bounds for $n_k$. We study the same quantity when $\omega (n)$ is replaced by $\Omega (n)$, the total number of prime factors of $n$ counted with repetitions.


References [Enhancements On Off] (What's this?)

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Ellipses, Paris, 2008.
  • Pál Erdős, Remarks on two problems, Mat. Lapok 11 (1960), 26–32 (Hungarian, with English and Russian summaries). MR 123538
  • P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292–301. MR 376517

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Additional Information

Jean-Marie De Koninck
Affiliation: Départment de Mathématiques, Université Laval, Québec G1K 7P4, Canada
MR Author ID: 55480
Email: jmdk@mat.ulaval.ca

John B. Friedlander
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Email: frdlndr@math.toronto.edu

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
MR Author ID: 630217
Email: fluca@matmor.unam.mx

Received by editor(s): May 16, 2008
Received by editor(s) in revised form: July 3, 2008
Published electronically: November 18, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society