consecutive p-smooth integers
- 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?
[Non-text portions of this message have been removed]
- --- In firstname.lastname@example.org, 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,
It was difficult getting just over 2 for the first time with
My only consolation is that the above is also good for 15 consecutive 15823-smooth integers.