In this post we consider positive integers $a$ of the form $$\pm x^2\pm N_k\tag{1}$$ with $x\geq 1$ integer and for integers $k\geq 1$ denoting $$N_k=\prod_{j=1}^k p_j$$ as the primorial of order $k$, thus here $p_j$ denotes the $j$th prime number.
Definitions. We denote this set of integers as $$\mathcal{A}=\{\text{positive integers }a=\pm x^2\pm N_k\text{ s.t. }x\geq 1 \text{ is integer and }N_k\text{ is a primorial for some }k\}.$$ For a fixed real $T>0$ we denote $\mathcal{A}(T)$ the set $$\mathcal{A}(T)=\mathcal{A}\cap[1,T],$$ and the counting function of these numbers $a$ in the segment $[1,T]$ as $\#\mathcal{A}(T)$.
Question.
Can you calculate a sharp upper bound on the counting function $\#\mathcal{A}(T)$?
Can you prove that $$\sum_{a\in \mathcal{A}}\frac{1}{a}$$ is divergent?
Many thanks.
I did few experiments to state previous conjectures. The sequence of integers with such representation starts as $1,2,3,5,6,7,10,11,14,15,18,19,\ldots$
To ask my questions I was inspired in similar statements that are in the literature Florian Luca and Alain Togbé, On numbers of the form $\pm x^2\pm y!$, from Diophantine Equations, Editor: N. Saradha, Narosa Publishing House, (2008).
I don't know if my sequence $(1)$ and the questions are in the literature, if it is in the literature answer the question as a reference request, and I try to find and read those propositions solving the Question.