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Denoting the number of primes in s_k by pi_k, we show that pi_k |s_k|/log p_{k+1}^2 and that pi(x) li(x) originates as a continuum approximation of the sum sum_k pi_k. In contrast, pi(x) x/log x stems from sieving repeatedly in regions already completed---explaining why x/log x underestimates pi(x). 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