Representing integers as linear combinations of power products

TL;DR

This paper investigates the properties and growth rate of a function related to representing integers as sums of power products of primes, providing new bounds and insights into additive number theory.

Contribution

It derives new results on the function F(k) and its variants, advancing understanding of integer representations using prime power products.

Findings

Bounds on the growth rate of F(k)

Properties of the related non-representable integers

Insights into additive representations involving prime powers

Abstract

Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A. In a recent paper Nathanson asked to determine the properties of the function F(k), in particular to estimate its growth rate. In this paper we derive several results on F(k) and on the related function which denotes the smallest positive integer which cannot be presented as sum of less than k terms from the union of A and -A.