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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 $g_k$ $(k \in \Bbb{N})$, defined by $$g_k(n) := \frac{n(n+1)\dots (n+k)} {\text{lcm}(n, n+1, \dots, n+k)}\qquad (\forall n \in \Bbb{N} \setminus \{0\}).$$ He proved that for each $k\in \Bbb{N}$, $g_k$ 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, {\it S. Hong} and {\it 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 $\text{lcm}(1, 2, \dots, k)$. In addition, they conjectured that $P_k$ is always a multiple of the positive integer $ \frac{\text{lcm}(1, 2, \dots, 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 $\text{lcm}(1, 2, \dots, k)$. In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $P_k$ $(k\in \Bbb{N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $\text{lcm}(1, 2, \dots, k)$ not divisible by some prime.

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