Is there a lower bound for the first non-trivial sequence of consecutive integers where each of the first $n$ primes is a least prime factor

Question

Using the Chinese Remainder Theorem, it is very straight forward to find a sequence of consecutive integers starting at $x$ where each of the first $n$ prime numbers is a least prime factor for a given number in the sequence and no number in the sequence has a least prime factor greater than $p_n$. Trivially, we know that this first occurs for the first $n$ primes at $x=2$. For example, the first $3$ primes are least prime factors in $2,3,4,5$

Each of these sequences can be characterized by the prime ordering of least prime factors which can be represented as $$p_1:p_2:\cdots:p_n$$

A sequence with the first $4$ primes is trivially found at $x=2$ in $2,3,4,5,6,7$. In this sequence, the prime ordering of the first occurrence of least prime factors is $2:3:5:7$. The second trivial occurrence is found at $x=3$ since $3,4,5,6,7$ shows the prime ordering $3:2:5:7$ A third trivial occurrence is found at $x=4$ since $4,5,6,7,8,9$ shows the prime ordering $2:5:7:3$.

I consider these sequences as "trivial" because for $n \ge 3$, $2$ such sequences at $x=2$ and $x=3$. There are multiple of these sequences in sequential order at $y=x+i$ since if $2:p_i:p_{i+1}:\cdots$ is an ordering so is $p_i:2:p_{i+1}:\cdots$. Depending on the number of primes, the same type of pattern will work for $3$ or any other of the smaller primes. For example, if $x$ is such a sequence and has the prime ordering of $3:2:5:7$, then necessarily, $x+1$ is such a sequence and has the prime ordering of $2:5:7:3$ at $x+1$

The first nontrivial occurrence for the $7$ is found at $x=90$ since $90, 91, 92, 93, 94, 95$ shows the prime ordering of $2:7:3:5$ I consider this nontrivial because $x > p_{n+1}$. There is always a long sequence of consecutive integers where no least prime factor is greater than $p_n$ between $2$ and $p_{n+1}-1$.

I know that there's been very interesting work with Jacobthal's function that details the upper bound for this type of sequence. Iwaniec has shown that

$$ j(n) \ll (\log n)^2. $$

By $j(n)$, I mean the Jacobthal functions which is defined as (from the OEIS wiki):

The ordinary Jacobsthal function j(n) is defined as the smallest positive integer m, such that every sequence of m consecutive integers contains an integer coprime to n.

Is there any paper or work that talks about the lower bound for the first nontrivial occurrence of a sequence of consecutive integers where each of the first $n$ primes is a least prime factor of a given number in the sequence and no number in the sequence has a least prime factor greater than $p_n$? Or anything that talks about the ordering of the least prime factors in such a sequence?

Thanks,

-Larry

Edit: I made mistake in my example. I had meant to use $x=90$ which I have fixed above. Add another example for $4$. Try to explain better what I mean by "trivial" and added a definition for $j(n)$.

Answer 1

It was proved by Rankin in 1963 that there are infinitely many $n$ for which $$j(n) \geq (C+o(1) \frac{\log(n) \log_{2}(n) \log_{4}(n)}{\log^{2}_{3}(n)} $$ holds for some positive $C>0$. The value of $C$ has since been improved by Maier and Pomerance (1990) to $C=e^{\gamma} \times 1.3125...$ and Pintz (1997) to $C=2e^{\gamma}$ (where $\gamma$ is Euler's constant).

Answer 2

Your conditions seem to imply a search for a confluence of a sizable prime gap in which a not very smooth number (one with least prime factor of $p_n$) occurs. You can limit the search by looking "between the totients" of P_n, the nth primorial, so a computer could find quickly some examples. However, I suspect that not much is known about the distribution of the totients of P_n; I'd be surprised if your question has been studied.