# 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

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 $j(n)$.

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Why use $x=84$ when $x=4$ works? –  The Masked Avenger Dec 23 '13 at 16:40
Not clear to me how you distinguish trivial from non-trivial. Also, you introduce notation $j(n)$ without defining it. –  Gerry Myerson Dec 23 '13 at 16:47
In fact, with 89 in your example, I think it does not work at all. You might try a formal definition along with a few more examples. –  The Masked Avenger Dec 23 '13 at 16:48
Masked Avenger, you are right. It should be $x=90$. I added detail on why I consider $x=4$ as trivial. @Gerry Myerson, I added more details on distinguishing between trivial and non-trivial. Thanks for your comments! –  Larry Freeman Dec 23 '13 at 17:39

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).
Thanks very much for the reference! Would you know if there is any work which can prove a lower bound that is always true? For example, by my definition of "trivial", we know that the first nontrivial sequence for $p_n$ is found at $x > p_{n+1}$ –  Larry Freeman Dec 23 '13 at 17:49
Since you need a number with LPF $p_n$, the next example is likely larger than $p_n^2$. –  The Masked Avenger Dec 23 '13 at 17:57
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.
In spite of my suspicions, it seems clear that you want to look mod $P_n$ at the $\phi(P_{n-1})$ many multiples of $p_n$ to find your example. You may get some nice results from this if you can show one nontrivial such sequence exists. –  The Masked Avenger Dec 23 '13 at 18:07