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A165262 Sorted hypotenuses with no repeats of Primitive Pythagorean Triples (PPT) if sum of all 3 sides are averages of twin prime pairs.

%I A165262 #10 Jan 30 2021 01:36:20
%S A165262 5,13,85,113,145,197,221,241,349,457,541,569,625,821,829,841,1025,
%T A165262 1037,1093,1157,1241,1433,1465,1621,1741,1769,2029,2069,2249,2353,
%U A165262 2441,2465,2501,2669,2725,2801,2809,2825,2873,3029,3077,3221,3293,3305,3389,3889
%N A165262 Sorted hypotenuses with no repeats of Primitive Pythagorean Triples (PPT) if sum of all 3 sides are averages of twin prime pairs.
%e A165262 Triples begin 3,4,5; 5,12,13; 15,112,113; 21,220,221; 24,143,145; 28,195,197; 36,77,85; 41,840,841; 59,1740,1741; 64,1023,1025; 89,3960,3961; 100,2499,2501; ...
%e A165262 So with sorted hypotenuses:
%e A165262   3 +  4 +  5 = 12, and 11 and 13 are twin primes;
%e A165262   5 + 12 + 13 = 30, and 29 and 31 are twin primes; ...
%t A165262 amax=10^5; lst={}; k=0; q=12!; Do[If[(e=((n+1)^2-n^2))>amax,Break[]]; Do[If[GCD[m,n]==1,a=m^2-n^2; b=2*m*n; If[GCD[a,b]==1,If[a>b,{a,b}={b,a}]; If[a>amax,Break[]]; c=m^2+n^2; x=a+b+c; If[PrimeQ[x-1]&&PrimeQ[x+1],k++; AppendTo[lst,c]]]],{m,n+1,12!,2}],{n,1,q,1}]; Union@lst
%Y A165262 Cf. A009004, A020882, A020883, A165158, A165159, A165160, A165236, A165237, A165238, A165260, A165261.
%K A165262 nonn,uned
%O A165262 1,1
%A A165262 _Vladimir Joseph Stephan Orlovsky_, Sep 11 2009