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