Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations:
\begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 & 193 & 197 & 199 \\ 821 & 823 & 827 & 829 \end{array}
Using Mathematica I was able to move further, so the sequence $n_k$ starts with:
$$\{n_k\}=\{1,10,19,82,148,187,208,325,346,565,943,\dots\}$$
A question already exists on this topic, however there is not a lot of information there.
I would like to know if the sequence $n_k$ was studied before, and what can we tell about the distribution of $n_k$ among the natural numbers?
Distances $n_{k+1}-n_k$ seem to grow on average, but 'close' quadruples still exist even for large $n_k$, for example:
$$n_{872}=960055,~~~n_{873}=960058$$
The plot of all the distances for $n_k<10^6$ is provided below (there are $898$ of them):
As the author of the linked question stated, every $n_k$ has the form $3m+1$, so the distances are all divisible by $3$.
So, the main thing I ask is some reference on the topic, or additional information about this sequence.
Found OEIS A007811 with some information
It is conjectured that the number of such $n_k$ up to $X$ is of the size (ignoring leading constants) $$ \frac{X}{\log^4 X},$$ and similarly that the number of $k$-tuples up to $X$ in fixed, admissible configurations is of the size $$ \frac{X}{\log^k X}.$$
Your tuple is $(10n+1, 10n+3, 10n+7, 10n+9)$. If you were to consider the $8$-tuple $(10n+1, 10n+3, 10n+7, 10n+9, 10n+91, 10n+93, 10n+97, 10n+99)$, (which I think is admissible but I didn't actually check), then it is conjectured that the number of such $8$-tuples up to $X$ is of the size $$ \frac{X}{\log^8 X}.$$ Notice that this is actually two of your $4$-tuples separated by $90$. So conjecturally we believe there should be infinitely "smallish" gaps between $4$-tuples of your shape.
More distribution-style statements can be made along these lines. You'll get very far by looking up the prime $k$-tuple conjecture and studying its progress and results.