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A198440 Square root of second term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

%I A198440 #24 Oct 22 2021 11:34:15
%S A198440 5,13,17,25,29,37,41,61,53,65,65,85,73,85,89,101,113,97,109,125,145,
%T A198440 145,149,137,181,157,173,197,185,169,221,185,193,205,229,257,265,205,
%U A198440 221,233,241,269,313,265,293,325,277,317,281,365,289,305,305,365,401
%N A198440 Square root of second term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).
%C A198440 This sequence gives the hypotenuses of primitive Pythagorean triangles (with multiplicities) ordered according to nondecreasing values of the leg sums x+y (called w in the Zumkeller link, given by A198441). See the comment on the equivalence to primitive Pythagorean triangles in A198441. For the values of these hypotenuses ordered nondecreasingly see A020882. See also the triangle version A222946. - _Wolfdieter Lang_, May 23 2013
%H A198440 Ray Chandler, <a href="/A198440/b198440.txt">Table of n, a(n) for n = 1..10000</a>
%H A198440 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>
%H A198440 Reinhard Zumkeller, <a href="/A198435/a198435.txt">Table of initial values</a>
%F A198440 A198436(n) = a(n)^2; a(n) = A198389(A198409(n)).
%e A198440 From _Wolfdieter Lang_, May 22 2013: (Start)
%e A198440 Primitive Pythagorean triangle (x,y,z), even y, connection:
%e A198440 a(8) = 61 because the leg sum x+y = A198441(8) = 71 and due to A198439(8) = 49 one has y = (71+49)/2 = 60 is even, hence x = (71-49)/2 = 11 and z = sqrt(11^2 + 60^2) = 61. (End)
%t A198440 wmax = 1000;
%t A198440 triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
%t A198440 tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
%t A198440 DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]][[All, 2]] (* _Jean-François Alcover_, Oct 22 2021 *)
%o A198440 (Haskell)
%o A198440 a198440 n = a198440_list !! (n-1)
%o A198440 a198440_list = map a198389 a198409_list
%Y A198440 Cf. A020882, A222946.
%K A198440 nonn
%O A198440 1,1
%A A198440 _Reinhard Zumkeller_, Oct 25 2011