[QUOTE=LaurV;293177]You (or the guy behind of primefan.ru) should report that [URL="http://oeis.org/A002072/list"]here[/URL].[/QUOTE]The guy behind of primefan.ru is me :-)

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

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

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

For [TEX]k=1[/TEX], the case being discussed there, we have [TEX]\log{n} \approx 5.154\sqrt{p} + 7.276[/TEX]. Any thoughts about these constants?

27129807647978258459761875 * 27129807647978258459761876 is 157-smooth, according to [url]http://bbs.emath.ac.cn/thread-4652-1-1.html[/url]

683232593267939977798585217 * 683232593267939977798585218 is 173-smooth, according to [url]http://forum.index.hu/Article/showArticle?go=40456539&t=9087484[/url]

See also [url]http://doi.org/10.1090/S0025-5718-2010-02381-6[/url] and [url]http://dx.doi.org/10.1080/10586458.2013.768483[/url]