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Mathematicians have been interested in properties of abundant numbers – those which are smaller than the sum of their proper factors – for over 2,000 years. During the last century, one line of research has focused in particular on determining the density of abundant numbers in the integers. Current estimates have brought the upper and lower bounds on this density to within about 10−4, with a value of K ≈ 0.2476, but more precise values seem difficult to obtain. In this paper, we employ computational data and tools from inferential statistics to get more insight into this value. We also put a lower bound on the quantity of abundants in any interval of size 106. Finally, we consider the “time series” nature of our data, and consider the possibility of employing tools from this branch of statistics to more carefully refine our statistical estimates.
Klyve, D., Pidde, M., & Temple, K. E. (2019). Estimating the Density of the Abundant Numbers. Integers, 19(A16).
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This article was originally published in Integers. The full-text article from the publisher can be found here.