## consecutive p-smooth integers

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• An integer is called p-smooth for a given prime p, if its prime factors don t exceed p. Sometimes a few consecutive integers are p-smooth (of course, we are
Message 1 of 42 , Aug 1, 2011
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An integer is called p-smooth for a given prime p, if its prime factors don't exceed p. Sometimes a few consecutive integers are p-smooth (of course, we are interested in non-trivial cases, where the smallest integer in the chain exceeds the prime next to p): for example, 4374 and 4375 are 7-smooth, while 2430, 2431 and 2432 are 19-smooth. There are also some chains of larger length (all 41-smooth):
212380, 212381, 212382
1517, 1518, 1519, 1520, 1521
285, 286, 287, 288, 289, 290
Another six integers, from 3294850 to 3294855, are 239-smooth. Eight integers from 4895 to 4902 are 89-smooth, while 15 integers from 48503 ti 48517 are 379-smooth.

The case of two consecutive integers was already studied (Sloane's A002072, A145605), but there are little information about larger chains. It there any interesting results known?

Thanks,

Andrey

[Non-text portions of this message have been removed]
• ... Too hard! It was difficult getting just over 2 for the first time with log(287946949)/log(15823). My only consolation is that the above is also good for 15
Message 42 of 42 , Nov 8, 2011
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--- In primenumbers@yahoogroups.com, Andrey Kulsha <Andrey_601@...> wrote:
>
> >> http://www.primefan.ru/stuff/math/maxs.xls
> >> http://www.primefan.ru/stuff/math/maxs_plots.gif
> >
> > Thanks, Andrey. The gradients are fanning out better now:
>
> The files were updated again.
>
> Puzzle: find a chain of 13 consecutive p-smooth integers, starting at N,
> with log(N)/log(p) greater than
>
> log(8559986129664)/log(58393) = 2.71328
>
> Best regards,
>
> Andrey
>

Too hard!

It was difficult getting just over 2 for the first time with
log(287946949)/log(15823).

My only consolation is that the above is also good for 15 consecutive 15823-smooth integers.
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