2016/10/13 by Cohen, Peter, Cordwell, Katherine, Epstein, Alyssa +3
#11B83 #FOS: Mathematics #Number Theory (math.NT) #Primary 11A25 #Secondary 11N25
paper · doi:10.48550/arxiv.1610.04253
The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and Shevelev: k-near-perfect numbers. These are examples to the well-known pseudoperfect numbers first defined by Sierpiński, and are numbers such that the sum of all but at most k of its proper divisors equals the number. We establish their asymptotic order for all integers k≥ 4, as well as some properties of related quantities.