New results on the least common multiple of consecutive integers

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 $g_k(n) := \frac{n (n + 1) \dots (n + k)} {\operatorname{lcm}(n, n+1, \dots, n + k)}$ $(\forall n \in \mathbb{N} \setminus \{0\})$. He proved that for each $k \in \mathbb{N}$, $g_k$ 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 improved the period $k!$ of $g_k$ to $\operatorname{lcm}(1 , 2, \dots , k)$. In addition, they conjectured that $P_k$ is always a multiple of the positive integer $\frac{\operatorname{lcm}(1 , 2 , \dots , k , k + 1)}{k + 1}$. An immediate consequence of this conjecture is that if $(k + 1)$ is prime, then the exact period of $g_k$ is precisely equal to $\operatorname{lcm}(1 , 2 , \dots , k)$.

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