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

%I A121727 #21 Dec 30 2024 17:04:03
%S A121727 5,13,17,25,41,29,37,61,65,85,53,113,65,101,145,73,85,89,181,145,221,
%T A121727 97,125,109,197,265,149,313,257,173,137,365,185,157,325,421,229,169,
%U A121727 481,205,185,193,401,269,545,293
%N A121727 Hypotenuse of primitive Pythagorean triangles sorted on area (A024406), then on hypotenuse.
%C A121727 Complete triple (X,Y,Z), with X>Y>Z is given by X=a(n),Y=A121728(n),Z=A121729(n).
%H A121727 Robert Israel, <a href="/A121727/b121727.txt">Table of n, a(n) for n = 1..10000</a>
%p A121727 N:= 100000: # for triples with area <= N
%p A121727 R:= NULL:
%p A121727 for n from 1 while (2*n+1)*(n+1)*n <= N do
%p A121727   for m from n+1 by 2 while (m^2 - n^2)*m*n <= N do
%p A121727     if igcd(m,n) = 1 then
%p A121727       R:= R, [m^2-n^2,2*m*n,m^2+n^2,(m^2-n^2)*m*n]
%p A121727     fi
%p A121727 od od:
%p A121727 R:= sort([R], (s,t) -> s[4] < t[4] or (s[4] = t[4] and s[3] < t[3])):
%p A121727 R[..,3]; # _Robert Israel_, Dec 30 2024
%o A121727 (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 A121727 Cf. A024406, A121728, A121729.
%K A121727 nonn
%O A121727 1,1
%A A121727 _Lekraj Beedassy_, Aug 17 2006
%E A121727 a(20)=145 corrected by _Philippe Guglielmetti_, Dec 14 2016
%E A121727 a(43)=401 inserted by _Michael Somos_, Dec 21 2016