Tight upper and lower bounds for the reciprocal sum of Proth primes

Abstract Computing the reciprocal sum of sparse integer sequences with tight upper and lower bounds is far from trivial. In case Carmichael numbers or twin primes even first decimal digit unknown. For accurate exact structure needs to be unfolded. this paper we present explicit for reciprocals Proth nine precision. We show closed formulae calculating n th number $$F_n$$ F n , up numbers. give an efficiently computable analytic expression linear order convergence involving $$\Psi $$ Ψ function (the logarithmic derivative gamma function). disprove two conjectures Zhi-Wei Sun regarding distribution primes.