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 A306658 #47 Apr 02 2019 03:26:28
%S A306658 3,6,7,5,5,5,4,4,2,0,1,9,2,6
%N A306658 Decimal expansion of the constant S_1 = Sum_{j>=1} prime(2*j - 1)!/prime(2*j)!.
%C A306658 Derived from an open problem published in 2012, the constant S_1 is connected to the gap between the j-th and (j+1)-th primes.
%C A306658 Together with the constant S_2 (see A306700), S_1 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.
%C A306658 On Mar 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get:
%C A306658 0.367555442019264 < S_1 < 0.367555442019265 + Sum_{j>=2100000000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.367555442019265 + 1/1.94576*10^(-13) < 0.367555442019460.
%D A306658 M. Ripà, Congetture su interrogativi inediti: tra speculazioni, voli pindarici e riflessioni spicciole, Jun 2012. ISBN 9788863699463
%H A306658 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rosser%27s_theorem">Rosser's theorem</a>
%F A306658 Sum_{j>=1} prime(2*j - 1)!/prime(2*j)! = 1/3 + Sum_{j>=2} 1/(Product{k=prime(2*j - 1) + 1, prime(2*j)} k) = 1/3 + 1/(7*6) + 1/(13*12) + ...
%e A306658 S_1 = 0.36755544201926...
%t A306658 a = 0; Do[f = Prime[Range[n - 999999, n]]; Do[a += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[a, 100]], {n, 1000000, 100000000, 1000000}]; a
%o A306658 (PARI) suminf(j=1, prime(2*j - 1)!/prime(2*j)!) \\ _Michel Marcus_, Apr 02 2019
%Y A306658 Cf. A000040, A306700.
%K A306658 cons,nonn,more
%O A306658 0,1
%A A306658 _Marco Ripà_, Mar 03 2019