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 A306700 #22 Apr 02 2019 03:32:27
%S A306700 0,5,1,6,6,6,6,2,2,8,8,4,2
%N A306700 Decimal expansion of the constant S_2 = Sum_{j>=1} prime(2*j)!/prime(2*j + 1)!.
%C A306700 The constant S_2 is connected to the gap between the j-th and (j+1)-th primes.
%C A306700 Together with the constant S_1 (see A306658), S_2 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.
%C A306700 On Mar 07 2019, the first 2445000000 prime numbers were used and from Rosser's theorem we obtain:
%C A306700 0.05166662288423 < S_2 < 0.05166662288424 + Sum_{j>=1222500000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.05166662288424 + 3.22757*10^(-13) < 0.05166662288457.
%H A306700 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rosser%27s_theorem">Rosser's theorem</a>
%F A306700 Sum_{j>=1} prime(2*j)!/prime(2*j + 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j) + 1, prime(2*j + 1)} k) = 1/(5*4) + 1/(11*10*9*8) + 1/(17*16*15*14) + ...
%e A306700 S_2 = 0.0516666228842...
%t A306700 b = 0; Do[f = Prime[Range[n - 999999, n]]; Do[b += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[b, 100]], {n, 1000001, 100000001, 1000000}]; b
%o A306700 (PARI) suminf(j=1, prime(2*j)!/prime(2*j + 1)!) \\ _Michel Marcus_, Apr 02 2019
%Y A306700 Cf. A000040, A306658 (S_1), A306700, A306780.
%K A306700 cons,nonn,more
%O A306700 0,2
%A A306700 _Marco Ripà_, Mar 05 2019