TITLE: Sequence for which no closed form can exist
QUESTION [5 upvotes]: I was wondering whether there exists a (computable) sequence of numbers, for which it can be proven that no closed form can exist. 
Edit: By closed form I mean an expression involving only a constant number of elementary functions. So something like a sum can not occur in the expression.

REPLY [6 votes]: Inevitably, the answer will depend on what one means by closed form.  We consider computable sequences of non-negative integers, that is, computable functions  $f(x)$ from  the non-negative integers to the non-negative integers. We will allow the least number operator $\mu$, as well as operations of ordinary arithmetic. 
The least number operator may not be familiar, so we define it. Let $R(y,x_1,\dots,x_n)$ be a relation such that for all $x_1,\dots,x_n$ there is a $y$ such that $R(y, x_1,\dots,x_n)$ holds. Then $\mu y R(y,x_1,\dots,x_n)$ is the least such $y$. 
As a consequence of Matijasevic's solution of Hilbert's Tenth Problem, there is a fixed polynomial $P(e, x, u_1,\cdots, u_k)$ with the following property. 
For any computable function $f$, there is a non-negative integer $e=e(f)$ such that for any $x$,
$$f(x)=[\mu y( P(e, x, [y]_1,[y]_2, \dots, [y]_k)=0)]_0.$$
Here by $[w]_i$ we mean the exponent of the $i$-th prime $p_i$ in the prime power decomposition of $w$.  (The $0$-th prime is $2$.)
This gives what I would consider a positive answer to the closed form question: Every computable sequence has a closed form. Many theorems of the same general kind were known long before the work of Matijasevic, except that instead of a polynomial $P$, one had a more complicated function. 
If the $\mu$-operator is not allowed, there are quite a few different workarounds.

REPLY [4 votes]: The most famous and interesting one is probably the sequence of primes (if you mean a closed form in terms of elementary functions).
Goldbach proved that no polynomial with integer coefficients can give a prime for all integer values. However it is not fully clear that there isn't some elementary function that generates all primes.
There is a whole wikipedia article on formulas for primes.