Let $p_n$ be the $n$-th prime.

For each integer $k \ge 0$, do there exist an infinite number of $k+3$ consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+2+k})$ so that

(1) The gap between the 1st and 2nd, and between the 2nd and last, are equal: $p_{n+1}-p_n = p_{n+2+k}-p_{n+1}$.

(2) There are $k$ primes between the 2nd and last, i.e., between $p_{n+1}$ and $p_{n+2+k}$.

~~For $k=0$, the answer is ~~
Here are some examples:
*Yes* by the recent breakthroughs on prime gaps.

One could whimsically imagine "skipping" a flat stone on the primes,
where the first bounce covers the gap between
the 1st and 2nd primes, followed by $k+1$ smaller
bounces that
together cover the same gap before sinking on the last prime.

notfollow from results on progressions in primes (since nothing guarantees the primes to beconsecutive, which is also the point of my first comment) $\endgroup$6more comments