Definition:Superabundant Number

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Definition

Let $n \in \Z_{>0}$ be a positive integer.

Then $n$ is superabundant if and only if:

$\forall m \in \Z_{>0}, m < n: \dfrac {\map \sigma m} m < \dfrac {\map \sigma n} n$

where $\map \sigma n$ is the $\sigma$ function of $n$.


That is, if and only if $n$ has a higher abundancy index than any smaller positive integer.


Sequence

The sequence of superabundant numbers begins:

$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, \ldots$


Examples

4

The abundancy index of $4$ is:

$\dfrac {\sigma \left({4}\right)} 4 = \dfrac 7 4 = 1 \cdotp 75$


6

The abundancy index of $6$ is:

$\dfrac {\sigma \left({6}\right)} 6 = \dfrac {12} 6 = 2$


12

The abundancy index of $12$ is:

$\dfrac {\sigma \left({12}\right)} {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$


24

The abundancy index of $24$ is:

$\dfrac {\sigma \left({24}\right)} {24} = \dfrac {60} {24} = 2 \cdotp 5$


36

The abundancy index of $36$ is:

$\dfrac {\sigma \left({36}\right)} {36} = \dfrac {91} {36} = 2 \cdotp 52 \dot 7$


48

The abundancy index of $48$ is:

$\dfrac {\sigma \left({48}\right)} {48} = \dfrac {124} {48} = 2 \cdotp 58 \dot 3$


60

The abundancy index of $60$ is:

$\dfrac {\sigma \left({60}\right)} {60} = \dfrac {168} {60} = 2 \cdotp 8$


120

The abundancy index of $120$ is:

$\dfrac {\sigma \left({120}\right)} {120} = \dfrac {360} {120} = 3$


180

The abundancy index of $180$ is:

$\dfrac {\sigma \left({180}\right)} {180} = \dfrac {546} {180} = 3 \cdotp 0 \dot 3$


240

The abundancy index of $240$ is:

$\dfrac {\sigma \left({240}\right)} {240} = \dfrac {744} {240} = 3 \cdotp 1$


Also see

  • Results about superabundant numbers can be found here.


Sources