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Hypotenuse of primitive Pythagorean triangles sorted on area (A024406), then on hypotenuse.
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%I #21 Dec 30 2024 17:04:03

%S 5,13,17,25,41,29,37,61,65,85,53,113,65,101,145,73,85,89,181,145,221,

%T 97,125,109,197,265,149,313,257,173,137,365,185,157,325,421,229,169,

%U 481,205,185,193,401,269,545,293

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

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

%H Robert Israel, <a href="/A121727/b121727.txt">Table of n, a(n) for n = 1..10000</a>

%p N:= 100000: # for triples with area <= N

%p R:= NULL:

%p for n from 1 while (2*n+1)*(n+1)*n <= N do

%p for m from n+1 by 2 while (m^2 - n^2)*m*n <= N do

%p if igcd(m,n) = 1 then

%p R:= R, [m^2-n^2,2*m*n,m^2+n^2,(m^2-n^2)*m*n]

%p fi

%p od od:

%p R:= sort([R], (s,t) -> s[4] < t[4] or (s[4] = t[4] and s[3] < t[3])):

%p R[..,3]; # _Robert Israel_, Dec 30 2024

%o (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 */

%Y Cf. A024406, A121728, A121729.

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Aug 17 2006

%E a(20)=145 corrected by _Philippe Guglielmetti_, Dec 14 2016

%E a(43)=401 inserted by _Michael Somos_, Dec 21 2016