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 $3p$?