This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347310 #47 Sep 10 2024 08:44:11
%S A347310 3,8,19,43,100,236,562,1354,3300,8119,20136,50302,126451,319628,
%T A347310 811829,2070790,5302162,13621745,35101258,90696900,234924747,
%U A347310 609864582,1586430423,4134442382,10793331294,28221407514,73898377351
%N 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.
%C A347310 Suggested by Mertens's theorem that Sum_{p <= x} log(p)/p = log(x) + O(1).
%D A347310 Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Society, 2015. See page 16.
%F A347310 a(n) = pi(A347311(n)) = A000720(A347311(n)). - _Michel Marcus_, Sep 06 2021
%e A347310 a(1) = 3 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.
%t A347310 Table[k=1;While[Sum[Log@Prime@i/Prime@i,{i,++k}]<n];k,{n,8}] (* _Giorgos Kalogeropoulos_, Sep 08 2021 *)
%o A347310 (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
%Y A347310 Cf. A000720, A002387, A016088, A046024, A347311.
%K A347310 nonn,more
%O A347310 1,1
%A A347310 _N. J. A. Sloane_, Sep 06 2021
%E A347310 a(8)-a(16) from _Michel Marcus_, Sep 06 2021
%E A347310 a(17)-a(23) from _Jon E. Schoenfield_, Sep 06 2021
%E A347310 a(24)-a(27) from _Amiram Eldar_, Sep 10 2024