A230842 -id:A230842 - cp's OEIS frontend

A230841 Values of x such that x^2 + y^2 = 61^n with x and y coprime and 0 < x < y.

5, 11, 234, 1320, 9475, 117469, 266286, 9184560, 3302155, 520632300, 1387108806, 23922442439, 154165737965, 933420304380, 13338456688674, 22995028210081, 1026964091673115, 713853567388260, 54078566783400895, 171226928056302601, 2435077776719657394

Offset: 1

Colin Barker, Oct 31 2013

nonn

The corresponding y-values are in A230842.

			a(3)=234 because 234^2+415^2=226981=61^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A230842, A188948, A230622, A230644, A230710, A230712, A230743, A230759.

Table[Select[PowersRepresentations[61^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1, 1]], {n, 22}] (* T. D. Noe, Nov 04 2013 *)
Table[Select[PowersRepresentations[61^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

a(10) and a(20) corrected by Zak Seidov, Nov 02 2013

A230963 Values of y such that x^2 + y^2 = 73^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

8, 55, 549, 5280, 44403, 325008, 2685304, 27358559, 241709752, 1870181225, 12766175931, 138963670560, 1291487885997, 10519458225072, 74032715923371, 690521409218881, 6773980286782088, 57975621715535095, 433109386513469096, 3345582274543898400

Offset: 1

Colin Barker, Nov 02 2013

nonn

The corresponding x-values are in A230962.

			a(3)=549 because 296^2 + 549^2 = 389017 = 73^3.

Robert Israel, Table of n, a(n) for n = 1..1000
Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, and Oliver Riesen, On primitive solutions of the Diophantine equation x^2+ y^2= M, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).

Cf. A230962.

Cf. A188949, A230623, A230645, A230711, A230713, A230744, A230760, A230842.

f:=n ->  max([abs@Re,abs@Im]((3+8*I)^n)):
map(f, [$1..50]); # Robert Israel, Mar 31 2017

Table[Max[Abs[Re[(3 + 8I)^n]], Abs[Im[(3 + 8I)^n]]], {n, 30}] (* Indranil Ghosh, Mar 31 2017, after formula by Robert Israel *)

from sympy import I, re, im
print([max(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

From Robert Israel, Mar 31 2017: (Start)
a(n) = max(abs(Re((3+8i)^n)), abs(Im((3+8i)^n))).
a(n) = abs(Im(3+8i)^n) if and only if 1/4 < frac(n*arctan(8/3)/Pi) < 3/4.(End)

cp's OEIS Frontend

A230841 Values of x such that x^2 + y^2 = 61^n with x and y coprime and 0 < x < y.

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