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National and Regional Contests
PEN Problems
PEN J Problems
16
16
Part of
PEN J Problems
Problems
(1)
J 16
Source:
5/25/2007
We say that an integer
m
≥
1
m \ge 1
m
≥
1
is super-abundant if
σ
(
m
)
m
>
σ
(
k
)
k
\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}
m
σ
(
m
)
>
k
σ
(
k
)
for all
k
∈
{
1
,
2
,
⋯
,
m
−
1
}
k \in \{1, 2,\cdots, m-1 \}
k
∈
{
1
,
2
,
⋯
,
m
−
1
}
. Prove that there exists an infinite number of super-abundant numbers.
Divisor Functions