Given an integer $n\ge3$ and a bound $x$, I want to find all $n$-tuples $p_1<\ldots<p_n$ of primes such that $p_2-p_1=p_3-p_2=\cdots=p_n-p_{n-1}$ with $p_n\le x$. Of course $d=p_2-p_1$ must be a multiple of all the primes up to $n$ ($n\#$ or $\exp\vartheta(n)$).
I don't want to store each of these tuples, just enumerate them (to search for ones with interesting properties). Is there an efficient algorithm for generating these? The ranges I'm interested in at the moment are $14 \le n \le 17$ and $10^{10} \le x \le 10^{14}$ so efficiency is important.
I could break the range into intervals small enough to fit them in memory, then sieve for primes, then for each prime $p$ I can check differences $d=n\#,\ 2n\#,\ \ldots,\ \left\lfloor\frac{p}{(n-1)n\#}\right\rfloor n\#$ to see if any of them work. The sieving takes around $O(p)$ with proper optimizations and the checking takes around $O(\log^3p)$ for M-R (following up later with proofs is cheap, since only a few will actually be n-tuples; in principle M-R can be done in $O(\log^{2+\varepsilon}p)$ with fast multiplication but that's not really relevant in this range) for a total of $O\left(\left\lfloor\frac{p}{(n-1)n\#}\right\rfloor p\log^2p\right)$ which is quasi-quadratic in $p$. (You lose a log since primes are thin.) For the numbers I'm in the neighborhood of 10^18 to 10^27 operations, which is a lot -- two years on a 6-core 3 GHz CPU at the lower end, to a billion times longer on the high end. But I feel like there has to be a cleverer way to sieve that would be much better than this brute force approach.