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:
- $\forall m \in \Z_{>0}, m < n: \dfrac {\map {\sigma_1} m} m < \dfrac {\map {\sigma_1} n} n$
where $\sigma_1$ denotes the divisor sum function.
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:
- $\dfrac {\map {\sigma_1} 4} 4 = \dfrac 7 4 = 1 \cdotp 75$
6
The abundancy index of $6$ is:
- $\dfrac {\map {\sigma_1} 6} 6 = \dfrac {12} 6 = 2$
12
The abundancy index of $12$ is:
- $\dfrac {\map {\sigma_1} {12} } {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$
24
The abundancy index of $24$ is:
- $\dfrac {\map {\sigma_1} {24} } {24} = \dfrac {60} {24} = 2 \cdotp 5$
36
The abundancy index of $36$ is:
- $\dfrac {\map {\sigma_1} {36} } {36} = \dfrac {91} {36} = 2 \cdotp 52 \dot 7$
48
The abundancy index of $48$ is:
- $\dfrac {\map {\sigma_1} {48} } {48} = \dfrac {124} {48} = 2 \cdotp 58 \dot 3$
60
The abundancy index of $60$ is:
- $\dfrac {\map {\sigma_1} {60} } {60} = \dfrac {168} {60} = 2 \cdotp 8$
120
The abundancy index of $120$ is:
- $\dfrac {\map {\sigma_1} {120} } {120} = \dfrac {360} {120} = 3$
180
The abundancy index of $180$ is:
- $\dfrac {\map {\sigma_1} {180} } {180} = \dfrac {546} {180} = 3 \cdotp 0 \dot 3$
240
The abundancy index of $240$ is:
- $\dfrac {\map {\sigma_1} {240} } {240} = \dfrac {744} {240} = 3 \cdotp 1$
Also see
- Results about superabundant numbers can be found here.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$