nt.number theory - Why Is 163/ln(163) a Near-Integer?

What make us confident about some mystery in these observations?

1st note:
"An example discovered by Srinivasa Ramanujan around 1913 is $exp(pisqrt{163})$,
which is an integer to one part in $10^{30}$, and has second continued fraction term
$1,333,462,407,511$. (This particular example can be understood from the fact
that as $d$ increases $exp(pisqrt{d})$ becomes extremely close to
$j((1 + sqrt{-d})/2)$, which turns out to be an integer whenever there
is unique factorization of numbers of the form $a + b sqrt{-d}$ --- and $d=163$
is the largest of the 9 cases for which this is so.) Other less spectacular examples
include $e^{pi}-pi$ and $163/log(163)$."

2nd note:
"Any computation involving 163 gives an answer that is close to an integer:
$$
163pi = 512.07960dots, quad
163e = 443.07993dots, quad
163gamma = 94.08615dotstext{"}
$$
and
$$
text{"}67/log(67)=15.9345774031dots, quad
43/log(43)=11.432521184dots
$$
...nah, with class number 1 it's not connected.
It's just the same 163. $ddotsmile$"

A synthetic example of my own:
$$
root3of{163}-frac{49,163}{9,000}
=0.0000000157258dots
$$
(note the double appearance of 163).

So, let's feel that the prime 163 is a supernatural number. $ddotsmile$

EDIT. Another interpretation the original question is related to
the observation of Kevin O'Bryant who computed the first successive maxima
of the sequence $|n/log(n)|$ where $| cdot |$ denotes the distance
to the nearest integer. The existence of infinitely many terms
is guaranteed by the following

Problem.
For any $epsilon>0$, there exists an $n$ such that $|n/log(n)|<epsilon$.

See solution by Kevin Ventullo to this question. I hope that this fact demystifies the original problem in full.