New results on the least common multiple of consecutive integers

Authors:
Bakir Farhi and Daniel Kane

Journal:
Proc. Amer. Math. Soc. **137** (2009), 1933-1939

MSC (2000):
Primary 11A05

Published electronically:
December 29, 2008

MathSciNet review:
2480273

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Abstract | References | Similar Articles | Additional Information

Abstract: When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions , defined by . He proved that for each , is periodic and is a period of . He raised the open problem of determining the smallest positive period of . Very recently, S. Hong and Y. Yang improved the period of to . In addition, they conjectured that is always a multiple of the positive integer . An immediate consequence of this conjecture is that if is prime, then the exact period of is precisely equal to .

In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of . We deduce, as a corollary, that is equal to the part of not divisible by some prime.

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Additional Information

**Bakir Farhi**

Affiliation:
Département de Mathématiques, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

Email:
bakir.farhi@gmail.com

**Daniel Kane**

Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02139

Email:
aladkeenin@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-08-09730-X

Keywords:
Least common multiple,
arithmetic function,
exact period.

Received by editor(s):
July 28, 2008

Received by editor(s) in revised form:
August 17, 2008

Published electronically:
December 29, 2008

Communicated by:
Ken Ono

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.