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](http://oeis.org/wiki/Jacobsthal_function)):



> 
> 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 $j(n)$.



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
It was proved by [Rankin](http://www.ams.org/mathscinet-getitem?mr=160767) 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](http://www.ams.org/mathscinet-getitem?mr=972703) (1990) to $C=e^{\gamma} \times 1.3125...$ and [Pintz](http://www.ams.org/mathscinet-getitem?mr=1443763) (1997) to $C=2e^{\gamma}$ (where $\gamma$ is [Euler's constant](http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant)).


