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Israel Contests
Israel Olympic Revenge
2020 Israel Olympic Revenge
N
N
Part of
2020 Israel Olympic Revenge
Problems
(1)
Divisibility sequence equals identity infinitely many times.
Source: Israel Olympic Revenge 2020 SL N1.
5/3/2022
Let
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
be an infinite sequence of positive integers. Suppose that a sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
of positive integers satisfies
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
∑
n
≠
d
∣
n
a
d
a_{n}=\sum_{n\neq d|n}a_d
a
n
=
n
=
d
∣
n
∑
a
d
for every integer
n
>
1
n>1
n
>
1
. Prove that the exist infinitely many integers
k
k
k
such that
a
k
=
k
a_k=k
a
k
=
k
.
number theory
divisible