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A198439 Square root of first term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

%I A198439 #21 Oct 22 2021 11:34:07
%S A198439 1,7,7,17,1,23,31,49,17,47,23,71,7,41,41,79,97,7,31,73,127,119,89,17,
%T A198439 161,47,113,167,119,1,199,49,73,103,161,223,241,23,31,103,89,191,287,
%U A198439 151,217,287,137,233,71,337,79,137,17,281,359,391,49,113,119,217
%N A198439 Square root of first term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).
%C A198439 This sequence gives the values |x-y| of primitive Pythagorean triangles (x,y,z) with even y ordered according to the nondecreasing values of the leg sums x+y (called w in the Zumkeller link, and given in A198441). For the equivalence to primitive Pythagorean triples with even y see a comment in A198441. - _Wolfdieter Lang_, May 22 2013
%H A198439 Ray Chandler, <a href="/A198439/b198439.txt">Table of n, a(n) for n = 1..10000</a>
%H A198439 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>
%H A198439 Reinhard Zumkeller, <a href="/A198435/a198435.txt">Table of initial values</a>
%F A198439 A198435(n) = a(n)^2; a(n) = A198388(A198409(n)).
%e A198439 From _Wolfdieter Lang_, May 22 2013: (Start)
%e A198439 Primitive Pythagorean triple (x,y,z), y even, connection:
%e A198439 a(2) = 7 because the triple with second smallest leg sum x+y = 17 = A198441(2) is (5,12,13), and |x - y| = y - x = 12 - 5 = 7.
%e A198439 a(3) = 7 because x + y = A198441(3) = 23, (x,y,z) = (15,8,17) (the primitive triple with third smallest leg sum), and |x-y| = x - y = 15 - 8 = 7. (End)
%t A198439 wmax = 1000;
%t A198439 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 A198439 tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
%t A198439 DeleteCases[tt, t_List /; GCD@@t>1 && MemberQ[tt, t/GCD@@t]][[All, 1]] (* _Jean-François Alcover_, Oct 22 2021 *)
%o A198439 (Haskell)
%o A198439 a198439 n = a198439_list !! (n-1)
%o A198439 a198439_list = map a198388 a198409_list
%K A198439 nonn
%O A198439 1,2
%A A198439 _Reinhard Zumkeller_, Oct 25 2011