A347310 a(n) = smallest k such that Sum_{i=1..k} log(p_i)/p_i >= n, where p_i is the i-th prime.
3, 8, 19, 43, 100, 236, 562, 1354, 3300, 8119, 20136, 50302, 126451, 319628, 811829, 2070790, 5302162, 13621745, 35101258, 90696900, 234924747, 609864582, 1586430423, 4134442382, 10793331294, 28221407514, 73898377351
Offset: 1
Examples
a(1) = 3 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.
References
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Society, 2015. See page 16.
Programs
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Mathematica
Table[k=1;While[Sum[Log@Prime@i/Prime@i,{i,++k}] -
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
Formula
Extensions
a(8)-a(16) from Michel Marcus, Sep 06 2021
a(17)-a(23) from Jon E. Schoenfield, Sep 06 2021
a(24)-a(27) from Amiram Eldar, Sep 10 2024
Comments