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 A235912 #6 Jan 17 2014 03:15:02
%S A235912 0,0,0,0,0,0,0,0,0,1,0,2,3,2,4,2,6,5,6,7,4,8,7,8,8,11,7,12,9,9,12,5,
%T A235912 14,10,9,9,9,9,7,8,11,9,8,7,14,8,6,9,5,5,9,11,3,9,6,13,8,8,6,7,6,5,4,
%U A235912 3,1,5,5,5,6,5,7,7,4,7,11,8,3,5,3,10,4,4,3,9,2,4,4,5,8,12,13,4,9,5,11,5,12,7,4,4
%N A235912 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.
%C A235912 Conjecture: a(n) > 0 for all n > 11.
%C A235912 This implies that there are infinitely many odd primes p = 2*m + 1 with q = m*(m-1) - prime(m) and r = m*(m+1) - prime(m) both prime. Note that r - q = 2*m.
%H A235912 Zhi-Wei Sun, <a href="/A235912/b235912.txt">Table of n, a(n) for n = 1..10000</a>
%e A235912 a(10) = 1 since phi(5) + phi(5)/2 = 6 with 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 all prime.
%t A235912 PQ[n_]:=n>0&&PrimeQ[n]
%t A235912 p[n_]:=PrimeQ[2n+1]&&PQ[n(n-1)-Prime[n]]&&PQ[n(n+1)-Prime[n]]
%t A235912 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
%t A235912 a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
%t A235912 Table[a[n],{n,1,100}]
%Y A235912 Cf. A000010, A000040, A235592, A235728.
%K A235912 nonn
%O A235912 1,12
%A A235912 _Zhi-Wei Sun_, Jan 16 2014