I was thinking about the number of distinct least prime factors in a sequence of consecutive integers and I noticed:
That every $4$ integers, there are $3$ distinct least prime factors. Every $6$ integers, there are $4$ distinct least prime factors. Every $10$, integers, there are at least $5$ distinct prime factors.
The argument for this is based on grouping the sequence into $6x+i, 6x+i+1, \dots$. For a sequence of $4$ integers, at least one is either $6x+1$ or $6x+5$. For a sequence of $6$ integers, at least two are $6x+1$ and $6x+5$ and they will be relatively prime to each other. In the case of $10$, at least three will be $6x+1, 6x+5$ and all three will be relatively prime to each other.
The situation gets more complicated when I am looking for at least $6$ distinct least prime factors.
By dividing up integers into $30x+i, 30x+i+1, \dots$, I have found that there are at least $6$ distinct least prime factors in a sequence of $14$ integers. The argument for this consists of going through the $30$ different forms and verifying by hand that each one has at least $6$ distinct least prime factors.
Is this sequence of numbers well known? I did a search on the oeis of integers for: $1, 2, 4, 6, 10, 14, \dots$ and found A005574 that looked interesting but it is not obvious to me that it is a match.
Is this sequence well-known? Is there a standard method for identifying the next item or does it require brute force? Is what I am looking for an exact match with A005574?
Edit: I believe that I found an oeis sequence that is an exact fit.
The best match is A048670. I did not find this one originally because it starts with $2$.
For those interested, a paper on the computation of the Jacobsthal's function h(n) for n < 50 can be found here.