Which numbers are not the sum plus the product of three positive integers?

TL;DR

This paper studies the representation of numbers as the sum plus the product of three positive integers, providing bounds and conjecturing infinitely many numbers cannot be represented in this form.

Contribution

It introduces bounds for the number of representations and the count of non-representable numbers, and conjectures the infinitude of such non-representable numbers.

Findings

Average number of representations is about 0.5 log^2 n

Upper bounds established for R_3(n) and non-representable numbers

Conjecture that R_3(n)=0 infinitely often

Abstract

We investigate the number $R_3(n)$ of representations of $n$ as the sum plus the product of three positive integers. On average, $R_3(n)$ is $\frac{1}{2}\log^2 n$. We give an upper bound for $R_3(n)$ and an upper bound for the number of $n \leq N$ such that $R_3(n) = 0$. We conjecture that $R_3(n)= 0$ infinitely often.