I had previously asked a question on mathstack exchange (Conjecture on The Generalized Carmichael Numbers) concerning with a conjecture I had discovered. I worked on the problem for a long time and discovered one more conjecture.
With the same terms and notations explained in the previous question (Conjecture on The Generalized Carmichael Numbers) The new conjecture goes as follows: Let $k \geqslant 1$, we have
$$\lim_{n \to \infty}\dfrac{\mid C_{-k} \text{ }\cap \text{ } (0, X]\mid}{\mid C_{k} \text{ }\cap \text{ } (0, X]\mid - \left(\pi\left(\dfrac{X}{k}\right)\dfrac{\gcd(\lambda(k), k)}{\lambda(k)}\right)} = 1$$
where $\pi(X)$ is the prime-counting function. Also Note that
For $k \geqslant 1$, we have:
$C_k = $ Generalized Carmichael Numbers + $k\left(\text{primes} \equiv 1 \mod\left(\dfrac{\gcd(\lambda(k), k)}{\lambda(k)}\right)\right)$
For $k \leqslant 0$, we have $C_k$ = Generalized Carmichael Numbers.
I have worked on this conjecture for a long time but I'm not sure why the phenomenon is happening. I have used lot of brute force (computer program) for checking and verifying the conjecture for some large values and seems like it works.
Any help or advice on how to proceed further would be appreciated. I really appreciate your time and efforts. Thanks.
