Why Is $\frac{163}{\operatorname{ln}(163)}$ a Near-Integer? The $j$-function and the fact that 163 and 67 have class number 1 explain why:
$\operatorname{exp}(\pi\cdot \sqrt{163}) = 262537412640768743.99999999999925$,
$\operatorname{exp}(\pi\cdot \sqrt{67}) = 147197952743.9999987$.
But is there any explanation for these?:
$\frac{163}{\operatorname{ln}(163)} = 31.9999987 \approx 2^5$,
$\frac{67}{\operatorname{ln}(67)} = 15.93 \approx 2^4$,
$\frac{17}{\operatorname{ln}(17)} = 6.00025$.
These numbers seem too close to integers to occur by chance.
 A: On the other hand, Mathematica gives LogIntegral[163]=43.075210908806756346563...  and LogIntegral[67]=22.6520420103880266691324...  so this does not appear to be connected to x/Ln[x] in the context of the Prime Number Theorem
A: 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.
