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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: November 18, 2008
MathSciNet review: 2470816
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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.


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Additional Information

Jean-Marie De Koninck
Affiliation: Départment de Mathématiques, Université Laval, Québec G1K 7P4, Canada
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
Email: fluca@matmor.unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09702-5
PII: S 0002-9939(08)09702-5
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