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 A165238 #4 Feb 16 2025 08:33:11
%S A165238 29,65,293,485,785,1049,1469,1961,2105,3005,3725,3821,4145,4181,4685,
%T A165238 4745,5105,5501,6053,6929,6953,7121,7361,7841,8693,9029,9125,10025,
%U A165238 12041,12833,12965,13649,14285,14909,15173,15689,15773,15821,16493
%N A165238 Hypotenuses c of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.
%C A165238 Only one copy of c enters the sequence if multiple solutions exist, like with (a,b,c) = (3164,12573,12965) and (a,b,c) = (483,12956,12965).
%C A165238 Subsequence of A020882. [_R. J. Mathar_, Mar 25 2010]
%H A165238 Eric Weisstein, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>, MathWorld
%e A165238 (a,b,c) = (20,21,29), (33,56,65), (44,483,485), (56,783,785), (68,285,293), (273,4136,4145).
%e A165238 In the first case, for example, 2*20+1=41, 2*21+1 and 2*29+1 are all prime, which adds the half-hypotenuse 29 to the sequence.
%t A165238 amax=6*10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];
%t A165238 Do[If[GCD[m, n]==1,a=m^2-n^2;If[PrimeQ[2*a+1],b=2*m*n;If[PrimeQ[2*b+1],
%t A165238 If[GCD[a, b]==1,If[a>b,{a,b}={b,a}];If[a>amax,Break[]];c=m^2+n^2;
%t A165238 If[PrimeQ[2*c+1], k++;AppendTo[lst,c]]]]]];If[a>amax,Break[]],{m,n+1,12!,2}],{n,1,q,1}];Union@lst
%Y A165238 Cf. A020882, A020883, A165236, A165237
%K A165238 nonn
%O A165238 1,1
%A A165238 _Vladimir Joseph Stephan Orlovsky_, Sep 09 2009
%E A165238 Comments moved to examples and definition clarified by _R. J. Mathar_, Mar 25 2010