For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, I nevertheless hope that there is such a function that is **simple to explain**.


I have listed a few examples myself which I am not very satisfied with for various reasons:


1. *The halting function.* This function is one of the uncomputable functions (probably the only uncomputable function) that is most easy to explain.


My problem with this one is that it involves computability theory, and I do not want to draw in computability if that is not necessary.
2. *The busy beaver function.* Also an uncomputable function. Which is a bit harder to explain.
3. The function that is $1$ everywhere if Goldbach's conjecture is true, and $0$ everywhere if Goldbach's conjecture is false. 


Problem: this is a constant function.
4. The function that is $1$ if $n$ is even and every even number is the sum of $n$ primes, or $n$ is odd and every odd number is the sum of $n$ primes. (And $0$ in all other cases.) For $n=2$ I believe this is the Goldbach conjecture, and for $n=3$ I believe this is the weak Goldbach conjecture. So this function is already a headeache for $n=2,3$, let alone for $n>3$.


Personally, this would be my favourite if it wasn't so baroque and over the top.
5. *The Collatz characteristic function.* $1$ if the Collatz sequence converges, $0$ if it does not.


Problem: too easy to verify on individual arguments. And it has little illustrative value because the Collatz characteristic function seems to be $1$ everywhere. (Emphasis on "seems".)
6. The function $\mathbb{N}\to\mathbb{N}\_0:n\mapsto \mbox{number of living people 
aged}(n)$.


Problem: depends on the real world. Not really a pure Platonic math function. (And function values change constantly throughout time.)


BTW: My question rules out computable functions. Values of computable functions on their domain of definition are known due to, e.g., dovetailing. (So strictly speaking, the 3rd item should be ruled out for this reason since constant functions are computable.)


Thank you.



A function that is defined everywhere but has unknown values
How about something based on a generalized twin prime conjecture? That is, $f(n)=1$ if there are infinitely many pairs of primes that differ by $2n$ (and $f(n)=0$ if not).


**Added 9/15/12:** It might be worth modifying the example to say $f(n)=1$ if and only if there are infinitely many pairs of *consecutive* primes that differ by $2n$. That would bring it fully in line with Polignac's conjecture. It's worth noting (here, at least) that the function as originally proposed could conceivably still be identically 1 even if Polignac's conjecture were *false* for all $n>2$.


