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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.

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  • $\begingroup$ How many terms have you calculated? I found the sequence begins 1,2,4,6,10,14,22 so (if my calculation is correct) the sequence differs from A005574 after 14. $\endgroup$ – Matthew Conroy Apr 18 '17 at 5:44
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    $\begingroup$ I calculated 1,2,4,6,10,14,18 but I need to confirm. I don't think the sequence exists in oeis. I like your sequence because it is 1, 2*1, 2*2, 2*3, 2*5, 2*7, 2*11. :-) $\endgroup$ – Larry Freeman Apr 18 '17 at 5:54
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    $\begingroup$ The $18$ integers $200,\dots,217$ only have $6$ distinct smallest prime factors: $2, 3, 5, 7, 11$ and $211$. $\endgroup$ – Matthew Conroy Apr 18 '17 at 6:09
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    $\begingroup$ Thanks for the example! I see now the mistake that I made. I only considered the case where the first integer in the sequence was relatively prime to 30. I hadn't considered $2$ as the first element in the sequence. $\endgroup$ – Larry Freeman Apr 18 '17 at 6:24
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    $\begingroup$ I think that I found a matching OEIS sequence: A048669: Jacobsthal function applied to the product of the first $n$ primes. This seems to me to be an exact match to the list of distinct least prime factors. $\endgroup$ – Larry Freeman Apr 18 '17 at 20:48

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