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New results on the least common multiple of consecutive integers. (English) Zbl 1229.11007

Authors’ summary: When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions g k (k), defined by

g k (n):=n(n+1)(n+k) lcm(n,n+1,,n+k)(n{0})·

He proved that for each k, g k is periodic and k! is a period of g k . He raised the open problem of determining the smallest positive period P k of g k . Very recently, S. Hong and Y. Yang [C. R., Math., Acad. Sci. Paris 346, No. 13–14, 717–721 (2008; Zbl 1213.11014)] improved the period k! of g k to lcm(1,2,,k). In addition, they conjectured that P k is always a multiple of the positive integer lcm(1,2,,k,k+1) k+1. An immediate consequence of this conjecture is that if (k+1) is prime, then the exact period of g k is precisely equal to lcm(1,2,,k).

In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of P k (k). We deduce, as a corollary, that P k is equal to the part of lcm(1,2,,k) not divisible by some prime.


MSC:
11A25Arithmetic functions, etc.
11A05Multiplicative structure of the integers
11B83Special sequences of integers and polynomials