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2021-IMOC
N3
N3
Part of
2021-IMOC
Problems
(1)
sequence with greatest prime factor
Source: IMOC 2021 N3
8/12/2021
Define the function
f
:
N
>
1
→
N
>
1
f:\mathbb N_{>1}\to\mathbb N_{>1}
f
:
N
>
1
→
N
>
1
such that
f
(
x
)
f(x)
f
(
x
)
is the greatest prime factor of
x
x
x
. A sequence of positive integers
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
=
M
>
1
a_1=M>1
a
1
=
M
>
1
and
a
n
+
1
=
{
a
n
−
f
(
a
n
)
if
a
n
is composite.
a
n
+
k
otherwise.
a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases}
a
n
+
1
=
{
a
n
−
f
(
a
n
)
a
n
+
k
if
a
n
is composite.
otherwise.
Show that for any positive integers
M
,
k
M,k
M
,
k
, the sequence
{
a
n
}
\{a_n\}
{
a
n
}
is bounded.(TAN768092100853)
number theory
Sequences