superabundant number

English

Noun

superabundant number (plural superabundant numbers)

1. (number theory) 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, American Mathematical Society, 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 $1\leq m<n$. Let $Q(x)$ be the counting function of superabundant numbers. Then:

     a) If $n$ and $n'$ are two consecutive superabundant numbers then

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

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

   • 2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 151:

     It follows that there is a superabundant number in each real interval $[x,2x]$ and that the number of superabundant numbers is thus infinite.

2. Used other than figuratively or idiomatically: see superabundant,‎ number.

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

• (positive integer whose abundancy index is greater than that of any lesser positive integer): SA (abbreviation), superior highly composite number (Ramanujan)

Related terms

• abundant number
• highly abundant number
• superabundant

Translations

positive integer whose abundancy index is greater than that of any lesser positive integer

• Finnish: ylirunsas luku
• French: nombre superabondant m
• Romanian: please add this translation if you can
• Russian: please add this translation if you can

Further reading

• Colossally abundant number on Wikipedia.Wikipedia
• Highly composite number on Wikipedia.Wikipedia
• Superabundant Number on Wolfram MathWorld