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 A237769 #8 Feb 13 2014 04:58:12
%S A237769 0,0,0,0,0,0,0,0,1,2,2,3,2,2,3,3,3,4,3,4,4,2,2,2,2,4,4,2,2,2,2,3,3,1,
%T A237769 1,2,2,3,4,3,3,4,3,5,5,3,3,2,2,5,5,3,3,3,3,5,5,2,2,3,3,3,4,2,2,6,6,9,
%U A237769 8,4,4,3,3,6,6,5,5,4,4,7
%N A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.
%C A237769 Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.
%C A237769 (ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.
%C A237769 (iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.
%H A237769 Zhi-Wei Sun, <a href="/A237769/b237769.txt">Table of n, a(n) for n = 1..10000</a>
%e A237769 a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.
%e A237769 a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.
%e A237769 a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.
%t A237769 TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
%t A237769 a[n_]:=Sum[If[TQ[PrimePi[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
%t A237769 Table[a[n],{n,1,80}]
%Y A237769 Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.
%K A237769 nonn
%O A237769 1,10
%A A237769 _Zhi-Wei Sun_, Feb 13 2014