New results on the least common multiple of consecutive integers

Abstract

When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k \in \mathbb{N})$, defined by $g_k(n) := \frac{n (n + 1) ... (n + k)}{\lcm(n, n + 1, >..., n + k)}$ $(\forall n \in \mathbb{N} \setminus \{0\})$. He proved that $g_k$ $(k \in \mathbb{N})$ is periodic and $k!$ is a period of $g_k$. He raised the open problem consisting to determine the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang have improved the period $k!$ of $g_k$ to $\lcm(1, 2, ..., k)$. In addition, they have conjectured that $P_k$ is always a multiple of the positive integer $\frac{\lcm(1, 2, >..., k, k + 1)}{k + 1}$. An immediate consequence of this conjecture states 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 $P_k$ $(k \in \mathbb{N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $\lcm(1, 2, ..., k)$ not divisible by some prime.