A121727 Hypotenuse of primitive Pythagorean triangles sorted on area (A024406), then on hypotenuse.

5, 13, 17, 25, 41, 29, 37, 61, 65, 85, 53, 113, 65, 101, 145, 73, 85, 89, 181, 145, 221, 97, 125, 109, 197, 265, 149, 313, 257, 173, 137, 365, 185, 157, 325, 421, 229, 169, 481, 205, 185, 193, 401, 269, 545, 293

Offset: 1

Author

Lekraj Beedassy, Aug 17 2006

Keywords

• nonn

Comments

Complete triple (X,Y,Z), with X>Y>Z is given by X=a(n),Y=A121728(n),Z=A121729(n).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A024406, A121728, A121729.

Programs

Maple

N:= 100000: # for triples with area <= N
R:= NULL:
for n from 1 while (2*n+1)*(n+1)*n <= N do
  for m from n+1 by 2 while (m^2 - n^2)*m*n <= N do
    if igcd(m,n) = 1 then
      R:= R, [m^2-n^2,2*m*n,m^2+n^2,(m^2-n^2)*m*n]
    fi
od od:
R:= sort([R], (s,t) -> s[4] < t[4] or (s[4] = t[4] and s[3] < t[3])):
R[..,3]; # Robert Israel, Dec 30 2024

PARI

v=vector(M=10^4); for(a=1, M, v[a] = []; fordiv(2*a, x, if(x<(y=2*a/x) && issquare(x^2+y^2, &z) && 1==gcd([x,y,z]), v[a] = concat(z, v[a])))); concat(v) /* Michael Somos, Dec 21 2016 */

Extensions

a(20)=145 corrected by Philippe Guglielmetti, Dec 14 2016

a(43)=401 inserted by Michael Somos, Dec 21 2016