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Argentina Contests
Argentina National Olympiad
2022 Argentina National Olympiad
1
1
Part of
2022 Argentina National Olympiad
Problems
(1)
sum of max prime divisors of n such that P^k <=n
Source: 2022 Argentina OMA Finals L3 p1
3/25/2024
For every positive integer
n
n
n
,
P
(
n
)
P(n)
P
(
n
)
is defined as follows: For each prime divisor
p
p
p
of
n
n
n
is considered the largest integer
k
k
k
such that
p
k
≤
n
p^k\le n
p
k
≤
n
and all the
p
k
p^k
p
k
are added. For example, for
n
=
100
=
2
2
⋅
5
2
n=100=2^2 \cdot 5^2
n
=
100
=
2
2
⋅
5
2
, as
2
6
<
100
<
2
7
2^6<100<2^7
2
6
<
100
<
2
7
and
5
2
<
100
<
5
3
5^2<100<5^3
5
2
<
100
<
5
3
, it turns out that
P
(
100
)
=
2
6
+
5
2
=
89
P(100)=2^6+5^2=89
P
(
100
)
=
2
6
+
5
2
=
89
Prove that there are infinitely many positive integers
n
n
n
such that
P
(
n
)
>
n
P(n)>n
P
(
n
)
>
n
..
number theory