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"?*