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

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