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.



Why Is $\frac{163}{\operatorname{ln}(163)}$ a Near-Integer?
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


