#### Title

A Confidence Interval for the Density of Abundant Numbers

#### Document Type

Oral Presentation

#### Location

SURC Room 202

#### Start Date

15-5-2014

#### End Date

15-5-2014

#### Keywords

Statistics, Density, Abundant numbers

#### Abstract

For more than 2,000 years, mathematicians have studied the “sum of divisors” function σ(n). To give an example, since the divisors of 12 are 1, 2, 3, 4, 6 and 12, the sum of the divisors of 12 is σ(12) = 1+2+3+4+6+12 = 28. Depending on the value of σ(n) we can classify an integer n as deficient, perfect or abundant. An integer n is said to be perfect if σ(n) = 2n, deficient if σ(n) < 2n, and abundant if σ(n) > 2n. Notice this makes 12 an abundant number since σ(12) = 28 > 2(12). We notate the proportion of integers which are abundant numbers as dA. The current bounds on the proportion of abundant numbers are .2476171 < dA < .2476475. This presentation will cover how I used statistics to close the gap and give a new estimation of the proportion of abundant numbers.

#### Recommended Citation

Pidde, Melissa, "A Confidence Interval for the Density of Abundant Numbers" (2014). *Symposium Of University Research and Creative Expression (SOURCE)*. 7.

https://digitalcommons.cwu.edu/source/2014/oralpresentations/7

#### Additional Mentoring Department

Mathematics

A Confidence Interval for the Density of Abundant Numbers

SURC Room 202

For more than 2,000 years, mathematicians have studied the “sum of divisors” function σ(n). To give an example, since the divisors of 12 are 1, 2, 3, 4, 6 and 12, the sum of the divisors of 12 is σ(12) = 1+2+3+4+6+12 = 28. Depending on the value of σ(n) we can classify an integer n as deficient, perfect or abundant. An integer n is said to be perfect if σ(n) = 2n, deficient if σ(n) < 2n, and abundant if σ(n) > 2n. Notice this makes 12 an abundant number since σ(12) = 28 > 2(12). We notate the proportion of integers which are abundant numbers as dA. The current bounds on the proportion of abundant numbers are .2476171 < dA < .2476475. This presentation will cover how I used statistics to close the gap and give a new estimation of the proportion of abundant numbers.

## Faculty Mentor(s)

Klyve, Dominic