The Density and Quantitative Upper Bound of Integers Not Expressible as the Sum of a Prime and a Zumkeller Number

A Zumkeller number is a positive integer whose set of positive divisors can be partitioned into two disjoint subsets of equal sums. In this paper, we investigate the additive properties of these numbers in conjunction with prime numbers. Specifically, we study the natural density of the exceptional set $\mathcal{E}$ consisting of all positive integers not expressible as the sum of a prime and a Zumkeller number. Let $\alpha (x)=\mathrm{\#}\{n\le x:n\in \mathcal{E}\}$ denote its counting function. We establish a quantitative upper bound for $\alpha (x)$, demonstrating that $\alpha (x)\ll x(\log \log x{)}^{-A}$, where $A={\log }_2\left(\frac{2{\pi }^2}{2{\pi }^2-3}\right)\approx 0.2378$, thereby proving that the natural density of this exceptional set is $0$.