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 A270992 #13 Jun 11 2017 18:16:30
%S A270992 0,1,1,1,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,2,
%T A270992 1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,2,1,1,1,1,1,2,1,1,1,1,1,1,
%U A270992 1,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1
%N A270992 Number of distinct prime divisors of prime(n)+1 and prime(n+1)+1.
%H A270992 Harvey P. Dale, <a href="/A270992/b270992.txt">Table of n, a(n) for n = 1..1000</a>
%e A270992 For n=1, p=2 and q=3; 3 and 4 have no common prime divisor, so a(1)=0.
%e A270992 For n=2, p=3 and q=5; 4 and 6 have 1 common prime divisor, so a(2)=1.
%e A270992 For n=9, p=23 and q=29; 24 and 30 have 2 common prime divisors, so a(9)=2.
%t A270992 Table[Length[Map[First, FactorInteger[GCD @@ {Prime@ n + 1, Prime[n + 1] + 1}]] /. 1 -> Nothing], {n, 101}] (* _Michael De Vlieger_, Mar 28 2016 *)
%t A270992 Length[Intersection[FactorInteger[#[[1]]+1][[All,1]],FactorInteger[#[[2]] + 1][[All,1]]]]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale_, Jun 11 2017 *)
%o A270992 (PARI) lista(nn) = {p = 2; f = factor(p+1)[,1]~; forprime(q=3, nn, g = factor(q+1)[,1]~; print1(#setintersect(f, g), ", "); p = q; f = g;);}
%o A270992 (PARI) a(n) = my(p = prime(n), q = nextprime(p+1)); #setintersect(factor(p+1)[,1]~, factor(q+1)[,1]~);
%Y A270992 Cf. A008335, A270592 (records).
%K A270992 nonn
%O A270992 1,9
%A A270992 _Michel Marcus_, Mar 28 2016