# The Abundancy Ratio, a Measure of Perfection

@article{Weiner2000TheAR, title={The Abundancy Ratio, a Measure of Perfection}, author={Paul Weiner}, journal={Mathematics Magazine}, year={2000}, volume={73}, pages={307 - 310} }

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 3. Thanks also to Chris Fisher, University of Regina, Canada, whose technique m1d own generalization of Van Aubel's theorem (first communicated to me via e-mail in 1998) in combination with Theorem 3, led to Theorems 4 and 5.

## 22 Citations

### The Abundancy Index and Feebly Amicable Numbers

- Mathematics
- 2021

This research explores the sum of divisors - σ ( n ) - and the abundancy index given by the function σ ( n ) n . We give a generalization of amicable pairs - feebly amicable pairs (also known as…

### A Circle-Stacking Theorem

- Mathematics
- 2003

1. R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comp. 57 (1991), 857-868. 2. G. H. Hardy and E. M. Wright, An Introduction to…

### Abundancy “Outlaws” of the Form σ(N)+t N

- Mathematics
- 2007

The abundancy index of a positive integer n is defined to be the rational number I(n) = �(n)/n, whereis the sum of divisors function �(n) = P d|n d. An abundancy outlaw is a rational number greater…

### Measuring Abundance with Abundancy Index

- Mathematics
- 2021

A positive integer n is called perfect if σ(n) = 2n, where σ(n) denote the sum of divisors of n. In this paper we study the ratio σ(n) n . We define the function Abundancy Index I : N → Q with I(n) =…

### A Simpler Dense Proof Regarding the Abundancy Index

- Philosophy, Economics
- 2003

(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…

### The Form Of The Friendly Number Of 10

- Mathematics
- 2020

Any positive integer n other than 10 with abundancy index 9/5 must be a square with atleast6 distinct prime factors, the smallest being 5, and my new argument about the form of the friendly number of…

### Does Ten Have a Friend

- Mathematics
- 2008

Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be…

### Characterizing Quasi-Friendly Divisors

- Mathematics
- 2020

Abundancy ratios are rational numbers k m satisfying σ(N) N = k m for some N ∈ Z≥1, where σ is the sum-of-divisors function. In this paper we examine abundancy ratios of the form σ(m)+1 m , where…

### Conditions Equivalent to the Existence of Odd Perfect Numbers

- Mathematics
- 2006

7. R. L. Duncan. A variation of the Buffon needle problem. this MAGAZINE, 40 (1967), 36–38. 8. H. J. Khamis. Buffon’s needle problem on radial lines. this MAGAZINE, 64 (1991), 56–58. 9. Daniel A.…

### Generalizations of the Abundancy Index and Outlaws

- Mathematics
- 2018

The abundancy index is defined as the sum of all the positive divisors of a positive integer $n$ divided by $n$. An abundancy outlaw is defined as a rational number $\frac{a}{b}$ greater than 1 that…

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