In this post I present a similar question that shows section **A19** of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from my question harmonizes with this.

For integers $k\geq 1$ we denote the primorial of order $k$ as $N_k$, that is

$$N_k=\prod_{r=1}^k p_r$$

where thus $p_r$ denotes the $r-th$ prime number. I add as reference that Wikipedia has the article *Primorial.*

In next paragraph $n\geq 1 $ denotes the integer variable.

**Definition.** *Now we consider the values of* $n$ *making* $n-N_k$ *prime for all those primorials* $N_k$ *satisfying* $2\leq N_k<n$.

I've calculated the first few terms from the definition. Our sequence starts as $$n=4,9,25,43,49,73\ldots...$$

To illustrate our sequence we add the example about why $25$ is in our sequence: the only primorials less than $n=25$ are $N_1=2$ and $N_2=6$, and we get that the integers $25-2$ and $25-6$ are both prime numbers.

I have no idea/intutition if this sequence has only a finite number of terms.

Question.I don't know if it exists only a finite number of positive integers $n's$ being $$n-N_k$$ a prime number for all primorial $2\leq N_k<n$. What work can be done to study if the sequence of $n's$ from this definition should have a finite number of terms?Many thanks.

I am asking if it is possible to deduce a proposition, get some idea about it, or set some conjecture.

**I emphasize** that I am asking about what work or statements/heuristics we can get for the **Question,** since it is a hard question (see the comments) I think that I should accept an available and useful answer.

## References:

[1] Richard K. Guy, *Unsolved Problems in Number Theory*, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).