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 A238701 #10 Mar 03 2014 10:16:58
%S A238701 0,0,0,0,0,0,0,0,0,1,2,2,3,3,4,4,4,3,3,3,3,4,4,4,6,5,5,5,3,4,6,6,7,6,
%T A238701 4,4,4,4,5,5,5,5,4,4,4,4,3,3,4,4,6,6,4,5,5,5,7,6,6,6,5,5,4,4,5,5,5,5,
%U A238701 5,5,6,6,5,5,5,5,3,4,5,5
%N A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.
%C A238701 Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.
%C A238701 In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.
%H A238701 Zhi-Wei Sun, <a href="/A238701/b238701.txt">Table of n, a(n) for n = 1..10000</a>
%H A238701 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A238701 a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.
%t A238701 PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
%t A238701 p[n_,k_]:=PQ[Floor[(n-Prime[k])/4]]
%t A238701 a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
%t A238701 Table[a[n],{n,1,80}]
%Y A238701 Cf. A000040, A045917, A046927, A049002, A062326, A230502, A237413, A237705.
%K A238701 nonn
%O A238701 1,11
%A A238701 _Zhi-Wei Sun_, Mar 03 2014