  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.
