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 A224481 #64 May 07 2023 06:31:05
%S A224481 8,26,46,58,74,76,82,92,134,164,236,248,304,314,362,368,394,416,454,
%T A224481 496,502,512,544,568,592,598,632,668,706,734,772,776,788,818,824,844,
%U A224481 898,944,986,1142,1184,1324,1328,1346,1426,1436,1462,1502,1522,1612,1766
%N A224481 Positive integers x such that x^2 - 34 is the average of a twin prime pair.
%C A224481 y = x^2 - 34 is one of a family of quadratics y = x^2 + c that produces averages of twin prime pairs. The first 24 negative numbers c that produce averages are congruent to either 0 or 2 (mod 6) (as calculated by maxima), and they differ by no more than 6. Other than that, I have not found an order to the sequence of negative numbers c. The first 11 positive numbers c that produce averages are apparently the beginning of all integers >= 2 that are equivalent to {2,0,2,0...} (mod 6).
%C A224481 If c=2, then the x that satisfy y = x^2 + c are A080149.
%C A224481 Apparently, there are infinitely many numbers c that produce twin prime averages. Here are some of them: (-84, -78, -76, -72, -70, -66, -64, -60, -58, -54, -52, -46, -42, -40, -36, -34, -30, -28, -22, -18, -16, -12, -6, -4, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32).
%C A224481 Dickson's conjecture implies that this sequence is infinite. Bateman-Horn-Stemmler gives conjectured growth. - _Charles R Greathouse IV_, Apr 10 2013
%H A224481 Michael G. Kaarhus, <a href="/A224481/b224481.txt">Table of n, a(n) for n = 1..10000</a>
%e A224481 26 is in this sequence, because 26^2 - 34 = 642, which is the average of the twin prime pair (641, 643).
%t A224481 nn=1000; av = Select[Prime[Range[PrimePi[nn^2]]], PrimeQ[# + 2] &] + 1; Select[Range[nn], MemberQ[av, #^2 - 34] &] (* _T. D. Noe_, Apr 09 2013 *)
%t A224481 nn = 2000; Select[Range[8, nn, 2], PrimeQ[p = #^2 - 35] && PrimeQ[p + 2] &] (* _Zak Seidov_, Apr 27 2013 *)
%t A224481 Select[Range[3,1800],AllTrue[#^2-{35,33},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 14 2020 *)
%o A224481 (PARI) is(n)=isprime(n^2-35)&&isprime(n^2-33) \\ _Charles R Greathouse IV_, Apr 10 2013
%Y A224481 Cf. A080149.
%K A224481 nonn
%O A224481 1,1
%A A224481 _Michael G. Kaarhus_, Apr 09 2013