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Density of the set of numbers whose sum of digits is prime

The following paper by Glyn Harman (Counting primes whose sum of digits is prime. J. … Integer Seq. 15 (2012), no. 2, Article 12.2.2, 7 pp) studies a more complicated situation: count primes $p \leq X$, where in addition the sum of digits (in a given base $b$) is also prime. …
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