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A230962 Values of x such that x^2 + y^2 = 73^n with x and y coprime and 0 < x < y.

%I A230962 #28 May 08 2021 08:36:32
%S A230962 3,48,296,721,10072,213785,1958709,7613760,21165597,894454032,
%T A230962 12278087704,59926173839,62518379032,3374316625735,58552907681096,
%U A230962 416603004343680,1261259807092797,10231862403603888,255781764375436389,2697529798981443601,11543491568219853608
%N A230962 Values of x such that x^2 + y^2 = 73^n with x and y coprime and 0 < x < y.
%C A230962 The corresponding y-values are in A230963.
%H A230962 Vincenzo Librandi, <a href="/A230962/b230962.txt">Table of n, a(n) for n = 1..200</a>
%H A230962 Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, and Oliver Riesen, <a href="https://people.math.ethz.ch/~halorenz/publications/pdf/Miniatur.pdf">On primitive solutions of the Diophantine equation x^2+ y^2= M</a>, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).
%F A230962 From _Robert Israel_, Mar 31 2017: (Start)
%F A230962 a(n) = min(abs(Re((3+8i)^n)), abs(Im((3+8i)^n))).
%F A230962 a(n) = abs(Re(3+8i)^n) if and only if 1/4 < frac(n*arctan(8/3)/Pi) < 3/4.
%F A230962 (End)
%e A230962 a(3) = 296 because 296^2 + 549^2 = 389017 = 73^3.
%p A230962 f:=n ->  min([abs@Re,abs@Im]((3+8*I)^n)):
%p A230962 map(f, [$1..50]); # _Robert Israel_, Mar 31 2017
%t A230962 Table[Select[PowersRepresentations[73^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* _Vincenzo Librandi_, Mar 02 2014 *)
%t A230962 Table[Min[Abs[Re[(3 + 8I)^n]], Abs[Im[(3 + 8I)^n]]], {n, 30}] (* _Indranil Ghosh_, Mar 31 2017, after formula by _Robert Israel_ *)
%o A230962 (Python)
%o A230962 from sympy import I, re, im
%o A230962 print([min(abs(re((3 + 8*I)**n)), abs(im((3 + 8*I)**n))) for n in range(1, 31)]) # _Indranil Ghosh_, Mar 31 2017, after formula by _Robert Israel_
%Y A230962 Cf. A230963, A188948, A230622, A230644, A230710, A230712, A230743, A230759, A230841.
%K A230962 nonn
%O A230962 1,1
%A A230962 _Colin Barker_, Nov 02 2013