A Simpler Dense Proof Regarding the Abundancy Index

  title={A Simpler Dense Proof Regarding the Abundancy Index},
  author={Richard F. Ryan},
  journal={Mathematics Magazine},
  pages={299 - 301}
  • R. F. Ryan
  • Published 1 October 2003
  • Philosophy, Economics
  • Mathematics Magazine
(B) If I (a) = r/s is in lowest terms, then s divides a. This follows since sa (a) = ra and gcd(r, s) = 1. (C) If I (a) = r/s is in lowest terms then r > ar(s). This follows from properties (B) and (A) since r/s = I (a) > I (s) = a(s)/s. (The condition that r and s be relatively prime is an important one! Note that 1(2) = 6/4 even though 6 r then r/s is not an abundancy index. 

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Acknowledgment. I am indebted to Hessel Pot from Woerden in the Netherlands who in a personal communication to me in 1997 pointed out the additional properties to Theorems 1 and 2, as well as Theorem

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An Introduction to the Theory of Numbers

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THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.

Measuring the abundancy of integers, this MAGAZINE

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Results concerning uniqueness for σ(x)/x = σ(pnqm )/(pnqm ) and related topics

  • Int. Math. J
  • 2002