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A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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: A130271 A139369 A151555 * A103606 A139794 A202819

Adjacent sequences:  A263725 A263726 A263727 * A263729 A263730 A263731

KEYWORD

nonn,tabf

AUTHOR

Colin Barker, Nov 20 2015

STATUS

approved

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Last modified July 20 22:51 EDT 2017. Contains 289629 sequences.