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A347310
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a(n) = smallest k such that Sum_{i=1..k} log(p_i)/p_i >= n, where p_i is the i-th prime.
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1
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3, 8, 19, 43, 100, 236, 562, 1354, 3300, 8119, 20136, 50302, 126451, 319628, 811829, 2070790, 5302162, 13621745, 35101258, 90696900, 234924747, 609864582, 1586430423
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OFFSET
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1,1
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COMMENTS
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Suggested by Mertens's theorem that Sum_{p <= x} log(p)/p = log(x) + O(1).
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REFERENCES
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Tenenbaum, G. (2015). Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. See page 16.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 3 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.
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MATHEMATICA
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PROG
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(PARI) a(n) = my(k=0, s=0, p=2); while (s < n, s += log(p)/p; k++; p = nextprime(p+1)); k; \\ Michel Marcus, Sep 06 2021
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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