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 A306780 #17 May 23 2019 08:34:37
%S A306780 3,1,5,8,8,8,8,1,9,3,5,0
%N A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).
%C A306780 The constant S_1 - S_2 is related to the prime gaps, since twin primes produce the largest terms of the algebraic sum compared with neighboring terms.
%F A306780 S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!) = Sum_{j>=2} (-1)^j/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 - 1/(4*5) + 1/(6*7) - 1/(8*9*10*11) + ...
%e A306780 S_1 - S_2 = 0.315888819350...
%o A306780 (PARI) sumalt(j=1, (-1)^(j+1)*(prime(j)!/prime(j + 1)!)) \\ _Michel Marcus_, Apr 02 2019 \\ Needs default(realprecision, 10^4) _Jinyuan Wang_, May 19 2019
%Y A306780 Cf. A000040, A306658 (S_1), A306700 (S_2), A306744 (S_1 + S_2).
%K A306780 cons,nonn,more
%O A306780 0,1
%A A306780 _Marco Ripà_ and _Aldo Roberto Pessolano_, Mar 09 2019