When is $n/\ln(n)$ close to an integer? As usual I expect to be critisised for "duplicating"
this question. But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my response) is a mathematics
question motivated by Kevin's (O'Bryant) comment to the earlier post.
Problem.
For any $\epsilon>0$, there exists an $n$ such that 
$\|n/\log(n)\|<\epsilon$ where $\|\ \cdot\ \|$ denotes the
distance to the nearest integer.
In spite of the simple formulation, it is likely that
the diophantine problem is open. I wonder whether it follows
from some known conjectures (for example, Schanuel's conjecture).
 A: If $f(x)=\frac{x}{\log x}$, then $f'(x)=\frac{1}{\log x} - \frac{1}{(\log x)^2}$, which tends to zero as $x\rightarrow \infty$. Choose some large real number $x$ for which $f(x)$ is integral. Then the value of $f$ on any integer near $x$ must be very close to integral.
A: 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.
