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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On strings of consecutive integers with a distinct number of prime factors
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by Jean-Marie De Koninck, John B. Friedlander and Florian Luca PDF
Proc. Amer. Math. Soc. 137 (2009), 1585-1592 Request permission


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
  • MR Author ID: 55480
  • Email:
  • John B. Friedlander
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
  • Email:
  • Florian Luca
  • Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
  • MR Author ID: 630217
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1585-1592
  • MSC (2000): Primary 11A25, 11N64
  • DOI:
  • MathSciNet review: 2470816