noncomputable functions I know that there exist functions such that no computer program can, given arbitrary input, produce the correct function value. There is nothing, however which would prohibit us from knowing the function value for certain specific inputs.
Suppose we have an uncomputable function $f$ defined on N and an infinite sequence of programs $p_1,p_2,p_3,..$ such that $p_n$ computes $f(n)$ no matter what it is given as input.
Since we could use this infinite sequence of programs to compute the function value for an arbitrary input, I am led to believe this sequence of programs cannot exist.
Thus, for any uncomputable function there must exist a particular element in the domain of the function such that its function value cannot ever be computed.
Is my reasoning valid?

Yes, you're right !
The main problem of uncomputable function is that at some point (for a specific $n$), we can't compute it.
Obviously, if we could, for each $n$, find the right program $p_n$ that computes $f(n)$, then $f$ would be computable. As Robert Israel mentioned, it does not prevent $p_n$ from existing. We just don't know how to find it.
Note that the "we can't compute it for that particular $n$" is related to the underlying mathematical theory you use to do your computation. It means that there is no proof that $p_n$ is the right program for $f(n)$. But uncomputability is still uncomputabillity, so if you manage to prove that $p_n$ is indeed the program that computes $f(n)$ for that particular $n$ in a particular (recursive) theory $T$, be sure there is some other $m>n$ such that you won't be able to find $p_m$ using $T$.
