Unless the voters want to weigh in themselves, we can only guess.
As I see it, the question is not at all well motivated, and there is no hint why anyone might think that there are infinitely many such numbers. Indeed, as far as I know we don't have even a single such number.
Setting context or motivation should be the norm for a community of researchers who are trying to garner interest in their problems so that others will be willing to think about them. Particularly in number theory, it's very easy to cook up (possibly hard) random-looking problems, and this question does seem a little contrived to me. Maybe you could explain the real background and why you got interested?
As I see it, the reason why your answer hasn't attracted more interest is that it's a long mass of implications without much of a conceptual guideline, to a question that is of no obvious interest in the first place, and it's not well-formatted which doesn't help. In short, people don't seem to care to wade through it; people often hope to take some insight home with them after reading an answer, and they upvote answers which are recognizably insightful.
Gerhard's answer, even if incomplete, has the virtue of giving some insight or conceptual explanation in the form of heuristic. He also raises the better question (I think): are there any such $n$?
To reiterate: this is all looking into a crystal ball. In a constructive spirit, please take it as giving suggestions for asking good questions and giving good answers.