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If $f(n)$ is a strictly increasing elementary function from the integers 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 itthe sum to be transcendental?

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If $f(n)$ is a strictly increasing elementary function from the integers 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 it?

Post Closed as "no longer relevant" by David Loeffler, Felipe Voloch, Gerry Myerson, Kevin O'Bryant, S. Carnahan
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