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Let $A = (a_1, a_2, \ldots, a_n)$ be the sequence of odd primes are less than or equal to a prime number $p$.

Let $C$ be the infinite ascending sequence of composite numbers that their factors are all in $A$.

Let $B$ be a sequence of $n$ consecutive numbers of $C$ such that $b_n - b_{n-1} = a_2 - a_1, \ b_{n-1} - b_{n-2} = a_3 - a_2, \ldots, \ b_2 - b_1 = a_n - a_{n-1}$.

For example, when $p=5,\ A=3,5, \ C= 9,15,25,27,45,75,81,..., \ B=25,27$.

Does there exist such a sequence $B$ when $p>5$?

Alternative question:

If there exist such a sequence $B$ when $p>5$, then $b_1$ must be greater than $3p$?

ref: see the informative comments of the same question on MSE.

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  • $\begingroup$ Have you made any progress for any $p$ other than $p=5$? Is there any $p$ for which you can prove impossibility? any $p$ for which you have found such $B$? $\endgroup$
    – Gerry Myerson
    Commented Dec 29, 2019 at 15:21
  • $\begingroup$ @Gerry Myerson I guess there's no such $B$ when $p>5$. See the informative comments of the same question on MSE,that's all: math.stackexchange.com/questions/3486259/… $\endgroup$
    – user49311
    Commented Dec 31, 2019 at 11:36
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    $\begingroup$ When you crosspost, please mention so and include a link when you make the post (not just in response to later questioning). $\endgroup$
    – LSpice
    Commented Dec 31, 2019 at 11:42
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    $\begingroup$ @LSpice Thanks, I see that and updated the content. $\endgroup$
    – user49311
    Commented Dec 31, 2019 at 13:49

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