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%I #39 Feb 27 2024 19:01:16
%S 0,1,2,3,5,12,9,46,119,120,122,597,828,2035,4449,6554,239,28560,56403,
%T 86158,145668,341525,246046,1315911,3369960,3455641,3627003,17021162,
%U 23161315,58317492,128629846,186118929,13651680,815616479,1590277918,2474152797,4241902555,9719139348,6712571031,37641223154,95420159401,99498527400,107655263398,485257533003
%N List of coprime pairs (x,y) such that x^2+y^2 = 13^n with 0<x<y.
%C For n>2, all other solutions (x,y) are divisible by 13, e.g., 26^2+39^2=13^3.
%H Robert Israel, <a href="/A158936/b158936.txt">Table of n, a(n) for n = 0..3581</a>
%F a(2*n) = min(|Re((3+2*i)^n)|, |Im((3+2*i)^n)|), a(2*n+1) = max(|Re((3+2*i)^n)|, |Im((3+2*i)^n)|). - _Robert Israel_, Feb 27 2024
%e n=1: 13^1=2^2+3^2, hence a(1)=2, a(2)=3,
%e n=2: 13^2=5^2+12^2, hence a(3)=5, a(4)=12.
%p f:= proc(n) local q;
%p q:= map(abs, [Re,Im]((2+3*I)^n));
%p op(sort(q))
%p end proc:
%p map(f, [$0..50]); # _Robert Israel_, Feb 27 2024
%t s={2,3};x=2;y=3;Do[A=3x+2y;If[Mod[A,13]==0,A=Abs[3x-2y];B=2x+3y,B=Abs[2x-3y]];x=A;If[A>B,x=B;y=A,y=B];s=Join[s,{x,y}],{20}];s
%t Table[Select[PowersRepresentations[13^n, 2, 2], CoprimeQ @@ # &][[1]], {n, 0, 21}] (* _T. D. Noe_, Apr 12 2011 *)
%Y Cf. A098122 for case x^2+y^2=5^n.
%Y Cf. A188948, A188949 for the values of x and y separately.
%Y Cf. A188982, A188983 for even and odd terms.
%K nonn,tabf
%O 0,3
%A _Zak Seidov_, Apr 10 2011
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