The authors derive explicit asymptotic series for 1/li_k(x) whose coefficients come from a combinatorial recurrence, generalizing Panaitopol's expansion from primes to prime k-tuples.
Hunter, Asymptotic Analysis and Singular Perturbation Theory , 2004
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Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals
The authors derive explicit asymptotic series for 1/li_k(x) whose coefficients come from a combinatorial recurrence, generalizing Panaitopol's expansion from primes to prime k-tuples.