cp's OEIS Frontend

A198441 Square root of third term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

%I A198441 #31 Oct 22 2021 11:34:21
%S A198441 7,17,23,31,41,47,49,71,73,79,89,97,103,113,119,119,127,137,151,161,
%T A198441 161,167,191,193,199,217,217,223,233,239,241,257,263,271,281,287,287,
%U A198441 289,311,313,329,329,337,343,353,359,367,383,391,391,401,409,431,433
%N A198441 Square root of third term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).
%C A198441 This sequence gives the sum of the two legs (catheti) x + y of primitive Pythagorean triangles (x,y,z) with y even and gcd(x,y) = 1, ordered nondecreasingly (with multiple entries). See A058529(n), n>=2, for the sequence without multiple entries. For the proof, put in the Zumkeller link w = x + y, v = z and u = abs(x - y). This works because w^2 - v^2 = v^2 - u^2, hence u^2 = 2*v^2 - w^2 = 2*z^2 - (x+y)^2 = 2*(x^2 + y^2) - (x+y)^2 = x^2 + y^2 - 2*x*y = (x-y)^2. The primitivity of the arithmetic progression triples follows from the one of the Pythagorean triples: gcd(u,w) = 1 follows from gcd(x,y) = 1, then gcd(u,v,w) = gcd(gcd(u,w),v) = 1. The converse can also be proved: given a primitive arithmetic progression triple (u,v,w), 1 <= u < v < w, gcd(u,v,w) = 1, the corresponding primitive Pythagorean triple with even y is ((w-u)/2,(w+u)/2,v) or ((w+u)/2,(w-u)/2,v), depending on whether (w+u)/2 is even or odd, respectively. - _Wolfdieter Lang_, May 22 2013
%C A198441 n appears A330174(n) times. - _Ray Chandler_, Feb 26 2020
%H A198441 Ray Chandler, <a href="/A198441/b198441.txt">Table of n, a(n) for n = 1..10000</a>
%H A198441 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>
%H A198441 Reinhard Zumkeller, <a href="/A198435/a198435.txt">Table of initial values</a>
%F A198441 A198437(n) = a(n)^2; a(n) = A198390(A198409(n)).
%e A198441 Primitive Pythagorean triangle connection: a(1) = 7 because (u,v,w) = (1,5,7) corresponds to the primitive Pythagorean triangle (x = (w-u)/2, y = (w+u)/2, z = v) = (3,4,5) with leg sum 3 + 4 = 7. - _Wolfdieter Lang_, May 23 2013
%t A198441 wmax = 1000;
%t A198441 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 A198441 tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
%t A198441 DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]][[All, 3]] (* _Jean-François Alcover_, Oct 22 2021 *)
%o A198441 (Haskell)
%o A198441 a198441 n = a198441_list !! (n-1)
%o A198441 a198441_list = map a198390 a198409_list
%Y A198441 Cf. A225949 (triangle version of leg sums).
%Y A198441 Cf. A198384, A198385, A198386.
%Y A198441 Cf. A058529, A198390, A198409, A198437, A330174.
%K A198441 nonn
%O A198441 1,1
%A A198441 _Reinhard Zumkeller_, Oct 25 2011