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How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes

This might be a "shameless plug" but i did recently "generalize Meissel-Lehmer to count sum of the powers of primes". This was supposed to give an exposition, so it might be of ...
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How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes

If you consult Lehmer's paper, the terms in question arise from $$P_2(x, a) = \sum_{p_a < p_i \le \frac x{p_i}} \sum_{p_i \le p_j \le \frac x{p_i}} 1 = \sum_{p_a < p_i \le b} \left\{ \pi\left(\...
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