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%I #57 May 05 2021 21:03:18
%S 4,12,15,21,24,35,40,45,55,56,60,63,72,77,80,84,91,99,105,112,117,120,
%T 132,140,143,144,153,156,165,168,171,176,180,187,195,208,209,220,221,
%U 224,231,240,247,252,253,255,260,264,272,273,275,285,288,299,304,308,312,323
%N Ordered long legs of primitive Pythagorean triangles.
%C Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A < B); sequence gives values of B, sorted.
%C Any term in this sequence is given by f(m,n) = 2*m*n or g(m,n) = m^2 - n^2 where m and n are any two positive integers, m > 1, n < m, the greatest common divisor of m and n is 1, m and n are not both odd; e.g., f(m,n) = f(2,1) = 2*2*1 = 4. - _Agola Kisira Odero_, Apr 29 2016
%C All terms are composite. - _Thomas Ordowski_, Mar 12 2017
%C a(1) is the only power of 2. - _Torlach Rush_, Nov 08 2019
%C The first term appearing twice is 420 = a(75) = a(76) = A024410(1). - _Giovanni Resta_, Nov 11 2019
%C From _Bernard Schott_, May 05 2021: (Start)
%C Also, ordered sides a of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.
%C Example: a(2) = 12, because the second triple is (12, 10, 15) with side a = 12, satisfying 2/12 = 1/10 + 1/15 and 15-12 < 10 < 15+12.
%C The first term appearing twice 420 corresponds to triples (420, 310, 651) and (420, 406, 435), the second one is 572 = a(101) = a(102) = A024410(2) and corresponds to triples (572, 407, 962) and (572, 455, 770). The terms that appear more than once in this sequence are in A024410.
%C For the corresponding primitive triples and miscellaneous properties and references, see A343891. (End)
%D V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.
%H Giovanni Resta, <a href="/A020883/b020883.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%p for a from 4 to 325 do
%p for b from floor(a/2)+1 to a-1 do
%p c := a*b/(2*b-a);
%p if c=floor(c) and igcd(a,b,c)=1 and c-b<a then print(a); end if;
%p end do;
%p end do; # _Bernard Schott_, May 05 2021
%Y Cf. A020882, A020884, A020885, A020886, A024354, A024410.
%Y Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).
%K nonn
%O 1,1
%A _Clark Kimberling_
%E Extended and corrected by _David W. Wilson_
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