If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is algebraic, or has a closed form expression in terms of elementary functions?

If not, can we prove the sum to be transcendental?