Definition:Abundancy Index

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Definition

Let $n$ be a positive integer.

Let $\map \sigma n$ be the sigma function of $n$.

That is, let $\map \sigma n$ be the sum of all positive divisors of $n$.

Then the abundancy index of $n$ is defined as $\dfrac {\map \sigma n} n$.


Sequence of Abundancy Indices

The sequence of abundancy indices for the first few positive integers is as follows.

The rational non-integer numbers have been rounded to $3$ decimal places.

$\begin{array} {|r|r|l|} \hline n & \map \sigma n & \dfrac {\map \sigma n} n \\ \hline 1 & 1 & 1 \\ 2 & 3 & 1 \cdotp 5 \\ 3 & 4 & 1 \cdotp 333 \\ 4 & 7 & 1 \cdotp 75 \\ 5 & 6 & 1 \cdotp 2 \\ 6 & 12 & 2 \\ 7 & 8 & 1 \cdotp 143 \\ 8 & 15 & 1 \cdotp 875 \\ 9 & 13 & 1 \cdotp 444 \\ 10 & 18 & 1 \cdotp 8 \\ 11 & 12 & 1 \cdotp 091 \\ 12 & 28 & 2 \cdotp 333 \\ 13 & 14 & 1 \cdotp 077 \\ 14 & 24 & 1 \cdotp 714 \\ 15 & 24 & 1 \cdotp 6\\ 16 & 31 & 1 \cdotp 938 \\ 17 & 18 & 1 \cdotp 059 \\ 18 & 39 & 2 \cdotp 167 \\ 19 & 20 & 1 \cdotp 053 \\ 20 & 42 & 2 \cdotp 1 \\ \hline \end{array} \qquad \begin{array} {|r|r|l|} \hline n & \map \sigma n & \dfrac {\map \sigma n} n \\ \hline 21 & 32 & 1 \cdotp 524 \\ 22 & 36 & 1 \cdotp 636 \\ 23 & 24 & 1 \cdotp 043 \\ 24 & 60 & 2 \cdotp 5 \\ 25 & 31 & 1 \cdotp 24 \\ 26 & 42 & 1 \cdotp 615 \\ 27 & 40 & 1 \cdotp 481 \\ 28 & 56 & 2 \\ 29 & 30 & 1 \cdotp 034 \\ 30 & 72 & 2 \cdotp 4 \\ 31 & 32 & 1 \cdotp 032 \\ 32 & 63 & 1 \cdotp 969 \\ 33 & 48 & 1 \cdotp 455 \\ 34 & 54 & 1 \cdotp 588 \\ 35 & 48 & 1 \cdotp 371 \\ 36 & 91 & 2 \cdotp 528 \\ 37 & 38 & 1 \cdotp 027 \\ 38 & 60 & 1 \cdotp 679 \\ 39 & 56 & 1 \cdotp 436 \\ 40 & 90 & 2 \cdotp 25 \\ \hline \end{array}$


Abundant Number

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

$n$ is abundant if and only if $\dfrac {\sigma \left({n}\right)} n > 2$.


Perfect Number

A perfect number $n$ is a (strictly) positive integer such that:

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

where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.


Deficient Number

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

$n$ is deficient if and only if:

$\dfrac {\sigma \left({n}\right)} n < 2$


Also see


Sources