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A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

%I A263728 #75 Feb 16 2025 08:33:27
%S A263728 3,4,5,5,12,13,7,24,25,8,15,17,9,40,41,11,60,61,12,35,37,13,84,85,15,
%T A263728 112,113,16,63,65,17,144,145,19,180,181,20,21,29,20,99,101,21,220,221,
%U A263728 23,264,265,24,143,145,25,312,313,27,364,365,28,45,53
%N A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.
%C A263728 a(3*k+1)*a(3*k+2) / (a(3*k+1)+a(3*k+2)+a(3*k+3)) is always an integer for k >= 0. Also note that a(3*k+1)*a(3*k+2)/2 is never a perfect square. - _Altug Alkan_, Apr 08 2016
%D A263728 H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.
%H A263728 Colin Barker, <a href="/A263728/b263728.txt">Table of n, a(n) for n = 1..10000</a>
%H A263728 Yoshinosuke Hirakawa and Hideki Matsumura, <a href="https://doi.org/10.1016/j.jnt.2018.07.007">A unique pair of triangles</a>, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.
%H A263728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%H A263728 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>
%e A263728 The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015
%p A263728 a:=[]; b:={}; M:=30;
%p A263728 for u from 2 to M do for v from 1 to u-1 do
%p A263728    if gcd(u,v)=1 and u+v mod 2 = 1 then t1:=u^2-v^2; t2:= 2*u*v; t3:=u^2+v^2;
%p A263728    w:=sort([t1,t2]); a:=[op(a), [op(w),t3]]; b:={ op(b), op(w), t3};
%p A263728    fi:
%p A263728 od: od:
%p A263728 a;
%p A263728 sort(a); # A263728
%p A263728 sort(b); # A016825 and A042965 (Maple code from _N. J. A. Sloane_, Dec 15 2015)
%o A263728 (PARI)
%o A263728 \\ Primitive Pythagorean triples (a,b,c) with a<b<c for given a.
%o A263728 ppt(a) = {
%o A263728   my(L=List(), b, c, d, g);
%o A263728   fordiv(a^2, d,
%o A263728     g=a^2\d;
%o A263728     if(d<=g && (d+g)%2==0,
%o A263728       c=(d+g)\2; b=g-c;
%o A263728       if(a<b && gcd(b, c)==1, listput(L, [a,b,c]))
%o A263728     )
%o A263728   );
%o A263728   vecsort(Vec(L),,2)
%o A263728 }
%o A263728 concat(concat(vector(50, n, ppt(n))))
%Y A263728 Cf. A008846, A020882, A046086, A046087, A103606, A139794, A156678, A156679.
%Y A263728 See also A016825, A042965.
%K A263728 nonn,tabf
%O A263728 1,1
%A A263728 _Colin Barker_, Nov 20 2015