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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.

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    $\begingroup$ 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. $\endgroup$
    – Peter
    Aug 12, 2020 at 14:49
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    $\begingroup$ According to my calculations, $$p_1=7\ 321\ 991\ 041$$ does the job for length $8$. Please check. $\endgroup$
    – Peter
    Aug 12, 2020 at 15:05
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    $\begingroup$ And for length $9$ , the prime $$p_1=363\ 500\ 177\ 041$$ should do the job. $\endgroup$
    – Peter
    Aug 12, 2020 at 15:24
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    $\begingroup$ And finally, this should give length $10$ : $$p_1=2\ 394\ 196\ 081\ 201$$ $\endgroup$
    – Peter
    Aug 12, 2020 at 15:39
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    $\begingroup$ A093553 gives the value of $p_1$ for a given value of $k$. $\endgroup$ Aug 13, 2020 at 1:50

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