superabundant number

superabundant number (plural superabundant numbers)

1. Used other than with a figurative or idiomatic meaning: see superabundant,‎ number.
2. (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 ${\displaystyle Q(X)}$ is the number of superabundant numbers ${\displaystyle \leqq X}$ , then for ${\displaystyle c<5/48,\ Q(X)\geqq (\log X)^{1+c}}$ for sufficiently large ${\displaystyle X}$ .

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

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

     a) If ${\displaystyle n}$ and ${\displaystyle n'}$ are two consecutive superabundant numbers then

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

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

   • 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 ${\displaystyle [x,2x]}$ and that the number of superabundant numbers is thus infinite.

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

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

• abundant number
• highly abundant number
• superabundant

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