Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

In order to say clearly, this number should given by a certain formula, such as $\sum\_{i=1}^\infty f(n)$ (or $\int\_0^\infty f(x)dx$)where $f(n)$ is a certain function so that we can calculate $f(n)$ for any given integer number $n$ (or real number $x$). Hence we avoid the answers like "the least even integer $N$ which makes Goldbach conjecture not true" or "the age when I get married".

Thanks in advance!