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Miklós Schweitzer
2018 Miklós Schweitzer
5
5
Part of
2018 Miklós Schweitzer
Problems
(1)
Sum of highest prime power divisors
Source: Miklós Schweitzer 2018 P5
11/10/2018
For every positive integer
n
n
n
, define
f
(
n
)
=
∑
p
∣
n
p
k
p
,
f(n)=\sum_{p\mid n}{p^{k_p}},
f
(
n
)
=
p
∣
n
∑
p
k
p
,
where the sum is taken over all positive prime divisors
p
p
p
of
n
n
n
, and
k
p
k_p
k
p
is the unique integer satisfying
p
k
p
⩽
n
<
p
k
p
+
1
.
p^{k_p}\leqslant n<p^{k_p+1}.
p
k
p
⩽
n
<
p
k
p
+
1
.
Find
lim sup
n
→
∞
f
(
n
)
log
log
n
n
log
n
.
\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .
n
→
∞
lim
sup
n
lo
g
n
f
(
n
)
lo
g
lo
g
n
.
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