TITLE: Non-computable but easily described arithmetical functions
QUESTION [15 upvotes]: I have read about the existence of functions of the kind described in the title in several places, but never seen an instance of them. Sorry if this is too much an elementary question to be posted here.

REPLY [11 votes]: Joel and John have given lots of important examples; here is a curiosity:
John Conway invented the Turing complete language FRACTRAN. In this language, a program is simply a a finite sequence of fractions $p_1/q_1$, $p_2/q_2$, ..., $p_r/q_r$. Given an integer $n$, FRACTRAN multiplies $n$ by the first $p_i/q_i$ for which $q_i$ divides $n$. It does this repeatedly until no such $i$ exists, then it halts.
For any FRACTRAN program $(p_1/q_1, \ldots, p_r/q_r)$, consider the function $f$ which returns $1$ if the program halts and $0$ if it does not. This function can be rewritten as a simple piecewise-linear recurrence:

$f(n)=1$ if none of the $q_i$ divides $n$. 
  If $q_i$ is the first $q_i$ which divides $n$, then $f(n)=f(n p_i/q_i)$.
  If $f(n)$ is not forced by the above conditions, then $f(n)=0$.

Now take a program for which it is undecidable, given an input $n$, whether or not it will halt. Would you agree that this is an easily described arithmetic function, if it were written out as a recurrence like this?