Can you back up to the beginning and help me to understand your notation?
What's the sequence for the even numbers? What's the sequence for the primes? I think if you can simply show a couple of examples, we can help sort out the notational issues.
It's not possible for anyone (well, for me, anyway) to follow your argument since the definition of f[x] was so garbled. I get that to each term of a sequence you assign a pair of numbers, but I'm unclear on what those numbers are. Then you start talking about 2N, sort of out of the blue, without defining N. And if you're not sure whether 0 is an even number, let's talk about that.
In other words, let's nail down the Intro before going forward.