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

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, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53

Offset

1,1

Comments

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

References

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.

Links

Colin Barker, Table of n, a(n) for n = 1..10000

Yoshinosuke Hirakawa and Hideki Matsumura, A unique pair of triangles, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.

Eric Weisstein's World of Mathematics, Pythagorean Triple

Wikipedia, Pythagorean triple

Example

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

Maple

a:=[]; b:={}; M:=30;
for u from 2 to M do for v from 1 to u-1 do
   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;
   w:=sort([t1, t2]); a:=[op(a), [op(w), t3]]; b:={ op(b), op(w), t3};
   fi:
od: od:
a;
sort(a); # A263728
sort(b); # A016825 and A042965 (Maple code from N. J. A. Sloane, Dec 15 2015)

Prog

(PARI)
\\ Primitive Pythagorean triples (a, b, c) with a<b<c for given a.
ppt(a) = {
  my(L=List(), b, c, d, g);
  fordiv(a^2, d,
    g=a^2\d;
    if(d<=g && (d+g)%2==0,
      c=(d+g)\2; b=g-c;
      if(a<b && gcd(b, c)==1, listput(L, [a, b, c]))
    )
  );
  vecsort(Vec(L), , 2)
}
concat(concat(vector(50, n, ppt(n))))

Crossrefs

Cf. A008846, A020882, A046086, A046087, A103606, A139794, A156678, A156679.

See also A016825, A042965.

Sequence in context: A139369 A370731 A151555 * A370760 A103606 A139794

Adjacent sequences: A263725 A263726 A263727 * A263729 A263730 A263731

Keyword

nonn,tabf

Author

Colin Barker, Nov 20 2015

Status

approved