# Number of "primal" sequences of consecutive numbers of the form $p_1, 2p_2, 3p_3,\dots$ for primes $p_1, p_2,\dots$

I am interested in the number of "primal" sequences of consecutive numbers of the form $$p_1, 2 p_2, 3 p_3,\ldots, k p_k$$ for primes $$p_1, p_2,\ldots, p_k$$.

For instance, there are $$56,157$$ sequences of the form $$p_1, 2 p_2 = p_1 + 1$$ for $$p_1 < 20,000,000$$ where $$p_1$$ and $$p_2$$ are both prime. I have found sequences up to length 7 of which the earliest begins at $$5,516,281$$.

Are there any results limiting the existence or number of such primal sequences? Is there even an infinity of pairs of the form $$p_1$$, $$2 p_2$$? Is there an upper bound to the length of such sequences? If not, then clearly there are an infinitude of sequences of each particular length.

I am a beginner here and have checked many posts about twin primes but not seen something like this, apologies if it is a trivial or answered question already.

• I think, the generalized Bunyakovsky conjecture implies infinite many primes with every length, but already the case $n=2$ with length $2$ is unknown. Almost surely do infinite many primes $p$ exist, such that $2p-1$ is prime as well. but noone could yet prove it. Aug 12, 2020 at 14:49
• According to my calculations, $$p_1=7\ 321\ 991\ 041$$ does the job for length $8$. Please check. Aug 12, 2020 at 15:05
• And for length $9$ , the prime $$p_1=363\ 500\ 177\ 041$$ should do the job. Aug 12, 2020 at 15:24
• And finally, this should give length $10$ : $$p_1=2\ 394\ 196\ 081\ 201$$ Aug 12, 2020 at 15:39
• A093553 gives the value of $p_1$ for a given value of $k$. Aug 13, 2020 at 1:50