Representing integers with sums of proper divisors

Abstract

A positive integer n is a practical number if every integer up to n can be expressed as a sum of distinct divisors of n. Schwab and Thompson generalized this notion by defining f-practical numbers: positive integers n for which every integer in a specified interval can be expressed as a sum of f(d)-values for distinct divisors d of n, where f is an arithmetic function. One of the examples they studied arises when considering f-practical numbers with f = s, where s denotes the sum-of-proper-divisors function. In this case, these integers are referred to as s-practical numbers. In this thesis, we continue the research of Schwab and Thompson on s-practical numbers. We discuss various methods for deriving an upper bound on the count of s-practical numbers less than or equal to x. Moreover, we work toward a lower bound for this counting function by classifying all s-practical numbers with precisely two prime factors and partially classifying s-practical numbers of the form np^m, where n is s-practical and p is a prime coprime to n.