This is a follow on question to the question posted by Peter (Can we always find a prime of the form $\ p$#$+q\ $?). Replacing his variable $q$ with $p_{n+1}$, his conjecture in that question can be formulated as:
For all $n$, and $1\le i\le n$, the set $\{p_i\#+p_{n+1}\}$ contains at least one prime number.
I have been playing around with numbers, calculating primorials, etc., to see if I could observe any patterns that might shed light on the question. So far, no insights.
What I have noticed is that for $n=1,2$, every member of the set $\{p_i\#+p_{n+1}\}$ is a prime, but for $n>2$ there are always some composite numbers in the set. I have only been able to look up to $n=13$. In some instances ($n=6,\ 12$) a large fraction of the elements in a set are prime.
My question is: Is there any $n>2$ for which the set $\{p_i\#+p_{n+1}\}$ (where ($1\le i\le n$)) contains only prime numbers?
Some comments: $p_{n+1}$ must be the smaller member of a pair of twin primes, and the smaller member of a pair of sexy primes, since $p_1\#+p_{n+1}$ and $p_2\#+p_{n+1}$ must be a prime ($p_1\#=2$ and $p_2\#=6$). So if the twin prime conjecture were false, there would be some $n_0$ above which there could be no examples of that set containing all primes. Also, the larger $n$ becomes, the more sums are required to be prime, in ranges of numbers where primes become less frequent, which makes me feel that sets containing all primes will become less probable as $n$ becomes very large. Nonetheless, I cannot discover a line of reasoning that says such sets are impossible. Perhaps someone with access to greater computing power can find an example of such a set, or someone with deeper insights into number theory may see a way to prove such sets are either necessary or impossible.