superabundant number

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English

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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 is the number of superabundant numbers , then for for sufficiently large .
    • 1995, József Sándor, Dragoslav S. Mitrinović, Borislav Crstici, Handbook of Number Theory I, Springer, page 111,
      1) A number is called superabundant if for all with . Let be the counting function of superabundant numbers. Then:
      a) If and are two consecutive superabundant numbers then
      Corollary. .
    • 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 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 is a superabundant number if for all positive integers (where denotes the sum of the divisors of ).

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