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China Northern MO
2019 China Northern MO
8
8
Part of
2019 China Northern MO
Problems
(1)
2019 CNMO P8
Source: 2019 China North MO
2/22/2020
For positive intenger
n
n
n
, define
f
(
n
)
f(n)
f
(
n
)
: the smallest positive intenger that does not divide
n
n
n
. Consider sequence
(
a
n
)
:
a
1
=
a
2
=
1
,
a
n
=
a
f
(
n
)
+
1
(
n
≥
3
)
(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)
(
a
n
)
:
a
1
=
a
2
=
1
,
a
n
=
a
f
(
n
)
+
1
(
n
≥
3
)
. For example,
a
3
=
a
2
+
1
=
2
,
a
4
=
a
3
+
1
=
3
a_3=a_2+1=2,a_4=a_3+1=3
a
3
=
a
2
+
1
=
2
,
a
4
=
a
3
+
1
=
3
. (a) Prove that there exists a positive intenger
C
C
C
, for any positive intenger
n
n
n
,
a
n
≤
C
a_n\leq C
a
n
≤
C
. (b) Are there positive intengers
M
M
M
and
T
T
T
, satisfying that for any positive intenger
n
≥
M
n\geq M
n
≥
M
,
a
n
=
a
n
+
T
a_n=a_{n+T}
a
n
=
a
n
+
T
.
number theory