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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 ${g}_{k}$ $\left(k\in ℕ\right)$, defined by

${g}_{k}\left(n\right):=\frac{n\left(n+1\right)\cdots \left(n+k\right)}{\text{lcm}\left(n,n+1,\cdots ,n+k\right)}\phantom{\rule{2.em}{0ex}}\left(\forall n\in ℕ\setminus \left\{0\right\}\right)·$

He proved that for each $k\in ℕ$, ${g}_{k}$ 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 $\text{lcm}\left(1,2,\cdots ,k\right)$. In addition, they conjectured that ${P}_{k}$ is always a multiple of the positive integer $\frac{\text{lcm}\left(1,2,\cdots ,k,k+1\right)}{k+1}$. An immediate consequence of this conjecture is that if $\left(k+1\right)$ is prime, then the exact period of ${g}_{k}$ is precisely equal to $\text{lcm}\left(1,2,\cdots ,k\right)$.

In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of ${P}_{k}$ $\left(k\in ℕ\right)$. We deduce, as a corollary, that ${P}_{k}$ is equal to the part of $\text{lcm}\left(1,2,\cdots ,k\right)$ not divisible by some prime.

##### MSC:
 11A25 Arithmetic functions, etc. 11A05 Multiplicative structure of the integers 11B83 Special sequences of integers and polynomials
##### Keywords:
least common multiple; arithmetic function; exact period