Superabundant number

English

Noun

1. Template:Lb A positive integer whose abundancy index is greater than that of any lesser positive integer.
   • 1984, Richard K. Guy (editor), Reviews in Number Theory 1973-83, Volume 4, Part 1, As printed in Mathematical Reviews, Template:W, page 173,

     The authors prove a theorem: If $Q(X)$ is the number of superabundant numbers $\leqq X$, then for $c<5/48,Q(X)\geqq (\log X{)}^{1+c}$ for sufficiently large $X$.

   • 1995, József Sándor, Dragoslav S. Mitrinović, Borislav Crstici, Handbook of Number Theory I, Springer, page 111,

     1) A number $n$ is called superabundant if $\frac{\sigma (n)}{n}>\frac{\sigma (m)}{m}$ for all $m$ with $1\le m<n$. Let $Q(x)$ be the counting function of superabundant numbers. Then:

     a) If $n$ and $n^{\prime }$ are two consecutive superabundant numbers then

     $\frac{n}{n^{\prime }}<1+c(\log \log n{)}^2/\log n$

     Corollary. $Q(x)\ge c\log x\log \log x/(\log \log \log x{)}^2$.

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Usage notes

• In mathematical terms, a positive integer $n$ is a superabundant number if $\frac{\sigma (m)}{m}<\frac{\sigma (n)}{n}$ for all positive integers $m<n$ (where $\sigma (n)$ denotes the sum of the divisors of $n$).

Synonyms

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Related terms

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Translations

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Further reading

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• Superabundant Number on Template:W