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

Brian Conrey; Neil Shah

paper

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

arXiv:2112.15551

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 $\frac{1}{2} lo g^{2} n$ . 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.