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Old 2012-06-26, 07:37   #45
XYYXF
 
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Jan 2005
Minsk, Belarus

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Quote:
Originally Posted by LaurV View Post
You (or the guy behind of primefan.ru) should report that here.
The guy behind of primefan.ru is me :-)

I already reported these results here:
http://oeis.org/A193943
http://oeis.org/A193944
http://oeis.org/A193945
http://oeis.org/A193946
http://oeis.org/A193947
http://oeis.org/A193948
http://oeis.org/A199407
http://oeis.org/A200566
http://oeis.org/A200567
http://oeis.org/A200568
http://oeis.org/A200569
http://oeis.org/A200570
http://oeis.org/A209837
http://oeis.org/A209838
http://oeis.org/A209839

There are numbers n such that p(n,k) < p(i,k) for all i > n, where p(n,k) is the largest prime factor of \prod\limits_{d=0}^k (n+d).

Note that log n grows nearly as \sqrt{p(n,k)} irrespective of k:
http://www.primefan.ru/stuff/math/maxs_plots.gif
(the plot is taken from the Excel file noted before)

For k=1, the case being discussed there, we have \log{n} \approx 5.154\sqrt{p} + 7.276. Any thoughts about these constants?
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Old 2016-12-30, 23:48   #46
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27129807647978258459761875 * 27129807647978258459761876 is 157-smooth, according to http://bbs.emath.ac.cn/thread-4652-1-1.html

683232593267939977798585217 * 683232593267939977798585218 is 173-smooth, according to http://forum.index.hu/Article/showAr...6539&t=9087484

See also http://doi.org/10.1090/S0025-5718-2010-02381-6 and http://dx.doi.org/10.1080/10586458.2013.768483
Last fiddled with by XYYXF on 2016-12-31 at 00:09
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