Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer. Let
![\sigma(n)](/media/m/5/4/5/545f4c18781e48a12a394b66a96134c2.png)
be the sum of the natural divisors
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(including
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
). We say that an integer
![m \geq 1](/media/m/3/a/f/3aff592800c660b7eb0387617043d71e.png)
is superabundant (P.Erdos,
![1944](/media/m/2/b/b/2bb0e1351a5099fbde1cae8596fa7596.png)
) if
![\forall k \in \{1, 2, \dots , m - 1 \}](/media/m/0/4/6/04601973ee1cdb3cadf9d867ecac4e5b.png)
,
![\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.](/media/m/9/c/7/9c78914859231a1361734f686e5ab8ca.png)
Prove that there exists an infinity of superabundant numbers.
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Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is superabundant (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$
Prove that there exists an infinity of superabundant numbers.