Hi Wadim, I decided to do a numerical run the same way that Kevin O'Bryant did for the previous question, but this time for the easiest version of my revised problem, find new record lows for $$ \parallel n / \log n \parallel \cdot \log n $$

       2      0.0794415
      17      0.000719936
     163      6.42582e-06
  715533      5.17294e-06
 1432276      3.30032e-06
 6517719      3.13803e-06
11523158      1.61843e-06
11985596      1.18403e-06
24102781      7.51947e-08

The results are pretty similar to what O'Bryant found. So I suggest there is still an infinite sequence of "champion" numbers. So I do not think there is any full analysis possible, but there might (who knows?) be a subsequence with an explicit Ramanujan style recipe for construction.

Well, let us see if the numbers stay nicely spaced in tabular form once I post this. The preview seems to think so.

Hi Wadim, I decided to do a numerical run the same way that Kevin O'Bryant did for the previous question, but this time for the easiest version of my revised problem, find new record lows for $$ \parallel n / \log n \parallel \cdot \log n $$

       2      0.0794415
      17      0.000719936
     163      6.42582e-06
  715533      5.17294e-06
 1432276      3.30032e-06
 6517719      3.13803e-06
11523158      1.61843e-06
11985596      1.18403e-06
24102781      7.51947e-08

The results are pretty similar to what O'Bryant found. I suspect there is an infinite sequence of "champion" numbers. I do not think any full analysis of these is possible (you mention Lambert $W$), but there might (who knows?) be a subsequence with an explicit Ramanujan style recipe for construction.

Meanwhile, this is from C++ using "double" type, I imagine the accuracy is good enough. Easy enough to confirm with arbitrary precision in GP-Pari.