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The Product of Exponents in the Factorization of Consecutive Integers

Part of: Multiplicative number theory

Published online by Cambridge University Press:  21 December 2009

Jean-Marie de Koninck  and
Florian Luca
Jean-Marie de Koninck
Affiliation:
Département de mathématiques, Université Laval, Québec G1K 7P4, Canada, E-mail: jmdk@mat.ulaval.ca
Florian Luca
Affiliation:
Mathematical Institute, UNAM, Ap. Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacán, Mexico, E-mail: fluca@matmor.unam.mx
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Abstract

For each integer n ≥ 2, let β(n) stand for the product of the exponents in the prime factorization of n. Given an arbitrary integer k ≥ 2, let nk be the smallest positive integer n such that β(n + 1) = β(n + 2) = … = β(n + k). We prove that there exist positive constants c1 and c2 such that, for all integers k ≥ 2,

MSC classification

Secondary: 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 59 - 65
Copyright
Copyright © University College London 2009

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References

1.Heath-Brown, D. R., The divisor function at consecutive integers. Mathematika 31 (1984), 141–149.CrossRefGoogle Scholar