By the input size, do you mean the size of the tape for the Turing Machine? (sorry if this is a stupid question, I don't know much theoretical CS)
If that's what you mean I don't think it matches up with the original definition since Turing Machines are supposed to have infinite memory (i.e. infinite tape length)
I couldn't find any references to these functions online, but I simply may not be looking hard enough.
You said that you wanted to "submit" these functions. (I'm going to assume that by that you mean that you want to publish them somewhere.) It's probably possible to submit this to a math journal, but you would probably have to find some sort of practical use for these functions, and make sure that your claims about them are supported by a valid mathematical proof. It would probably be more difficult than if you were actually at a university, but it's probably possible.
If you just want to let other people know about it, and maybe ask questions about it, you could probably just post to an online forum. Some ones I know: Mathematics Stack Exchange, Art of Problem Solving Community, Math Is Fun Forum (however this one is not as active as the first two)