What make us confident about some mystery in these observations?

1st note:
"An example discovered by Srinivasa Ramanujan around 1913 is $\exp(\pi\sqrt{163})$,
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 $\exp(\pi\sqrt{d})$ 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:
$$
163\pi = 512.07960\dots, \quad
163e = 443.07993\dots, \quad
163\gamma = 94.08615\dots\text{"}
$$
and
$$
\text{"}67/\log(67)=15.9345774031\dots, \quad
43/\log(43)=11.432521184\dots
$$
...nah, with class number 1 it's **not** connected.
It's just the same 163. $\ddot\smile$"

A synthetic example of my own:
$$
\root3\of{163}-\frac{49,163}{9,000}
=0.0000000157258\dots
$$
(note the double appearance of 163).

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

**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.