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 A286634 #19 May 16 2017 00:15:17
%S A286634 2,2,1,1,1,1,3,1,1,2,1,1,3,3,1,1,2,1,3,2,1,2,2,1,1,1,1,7,1,3,1,5,1,3,
%T A286634 3,2,1,3,1,5,1,2,1,3,6,1,1,1,3,1,5,3,3,1,1,1,2,1,5,7,2,1,1,7,3,5,1,2,
%U A286634 3,2,1,3,2,1,2,2,1,5,1,1,1,3,2,1,2,2,1,1,6,2,1,1,1,3,3,1,3,3,5,1
%N A286634 Numerator of the ratio of alternate consecutive prime gaps: Numerator ((prime(n + 3) - prime(n + 2))/(prime(n + 1) - prime(n))).
%F A286634 a(n) = numerator((prime(n + 3) - prime(n + 2))/(prime(n + 1) - prime(n))).
%F A286634 A000040(n+3) = 5 + Sum_{k=1..n} ((1+(-1)^k)*Product_{j=1..k}(a(j)/A285851(j))^((1+(-1)^j)/2) + ((1-(-1)^k)/2)*Product_{j=1..k}(a(j)/A285851(j))^((1-(-1)^j)/2)), for n>0.
%F A286634 A001223(n+2) = (1+(-1)^n)*Product_{j=1..n}(a(j)/A285851(j))^((1+(-1)^j)/2) - ((-1+(-1)^n)/2)*Product_{j=1..n}(a(j)/A285851(j))^((1-(-1)^j)/2), for n>0
%t A286634 Table[Numerator[(Prime[k+3]-Prime[k+2])/(Prime[k+1]-Prime[k])], {k, 100}]
%Y A286634 Cf. A285851 (denominators), A001223, A000040, A272863, A276309.
%K A286634 nonn,frac
%O A286634 1,1
%A A286634 _Andres Cicuttin_, May 11 2017