Given discriminant $d$ and j-function $j(\tau)$, I was looking at,
$$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$
which appears in Ramanujan-type pi formulas. Let $C_d$ be the odd prime factors of the constant term of the minimal polynomial for $F(\sqrt{-d})$. Then for prime $d>3$,
$$\begin{aligned}
C_{5} &= 5, 11, 19.\\
C_{7} &= 3, 7, 19.\\
C_{11} &=7, 11, 19, 43.\\
C_{13} &=3, 13, 43.\\
C_{17} &=17, 19, 43, 59, \color{red}{67}.\\
C_{19} &=3, 19, \color{red}{67}.\\
C_{23} &=3, 7, 11, 19, 23, 43, \color{red}{67}, 83.\\
C_{29} &=7, 23, 29, \color{red}{67}, 107.\\
C_{31} &=3, 11, 23, 31, 43.\\
C_{37} &=3, 7, 11, 37, \color{red}{67}, 139.\\
C_{41} &=23, 31, 41, 43, 83, 139, \color{blue}{163}.\\
C_{43} &=3, 7, 19, 43, \color{blue}{163}.\\
C_{47} &=3, 11, 19, 31, 43, 47, \color{red}{67}, 107, 139, \color{blue}{163}, 179.\\
C_{53} &=7, 11, 43, 53, 131, \color{blue}{163}, 211.\\
C_{59} &=3, 5, 11, 23, 31, 43, 47, 59, \color{red}{67}, 211, 227.\\
C_{61} &=3, 19, 47, 61, \color{blue}{163}.\\
C_{67} &=3, 7, 11, 31, 43, \color{red}{67}.\\
C_{71} &=5, 7, 11, 23, 47, 59, \color{red}{67}, 71, \color{blue}{163}, 283.\\
\vdots\\
C_{163} &=3, 7, 11, 19, 59, \color{red}{67}, 127, \color{blue}{163}, 211, 571, 643.\\
C_{167} &=3, 43, \color{red}{67}, 103, 131, 139, 151, \color{blue}{163}, 167, 227, 307,\dots 659.\\
\end{aligned}$$
and so on. Notice that the d with $C_d$ divisible by $163$ are the first few primes of Euler's prime-generating polynomial,
$$P_1(n) = n^2+n+41 = 41, 43, 47, 53, 61, 71, 83, 97,\dots$$
and the lesser known,
$$P_2(n) = 4n^2+163 = 163, 167, 179, 199,\dots$$
Similarly, the d with $C_d$ divisible by $67$ intersect with,
$$Q_1(n) = n^2+n+17 = 17, 19, 23, 29, 37, 47, 59, 73, 89,\dots$$
and,
$$Q_2(n) = 4n^2+67 = 67, 71, 83, 103,\dots$$
Q: Does anybody know the reason for this "numerology"?
Best Answer
"Numerology" such as you've observed is explained in the paper
which gives more generally the factorizations of the constant terms of the minimal polynomials of $j(\tau) - j(\tau')$ where $\tau,\tau'$ are quadratic imaginaries not equivalent under ${\rm PSL}_2({\bf Z})$. Your $j(\tau) - 1728$ is the special case $\tau' = i$.
Before seeking patterns in the appearance of factors such as $67$ and $163$, one might wonder why all the constant terms, which are roughly exponential in $\sqrt d$, factor into such small prime factors in the first place. The reason is that these are the primes $p$ for which the elliptic curve $E: y^2 = x^3 - x$, which has $j$-invariant $1728$, is also the reduction mod $p$ of a curve of invariant $j(\sqrt{-d}\,)$, and thus has an action of ${\bf Z}[\sqrt{-d}\,]$. Since $E$ already has an action of ${\bf Z}[i]$ [with $i$ acting by $(x,y) \mapsto (-x,iy)$], this makes $E$ supersingular. The condition that the endomorphism ring of a supersingular curve $E \bmod p$ accommodate both ${\bf Z}[i]$ and ${\bf Z}[\sqrt{-d}\,]$ comes down to the representability of $4d$ by the quadratic form $a^2+pb^2$. In particular $p < 4d$, which explains why all the prime factors are small. Your $Q_1$ and $Q_2$ are obtained by setting $b=1$ and $b=2$, but eventually higher $b$ arise too, e.g. you'll find $p=67$ among the factors of $C_{151}$ (for which the minimal polynomial has degree $7$) because $151 = \frac14 (1^2 + 67 \cdot 3^2)$, even though $151$ is not represented by either $n^2+n+17$ or $n^2+67$.