A230622 -id:A230622 - cp's OEIS frontend

A230623 Values of y such that x^2 + y^2 = 17^n with x and y coprime and 0 < x < y.

4, 15, 52, 240, 1121, 4888, 20047, 77280, 277441, 1093425, 5279468, 23647519, 99429196, 393425745, 1457109628, 4968639359, 24553864319, 113193708472, 488133974353, 1980778750800, 7547952442399, 26710380775592, 112605054449252

Offset: 1

Colin Barker, Oct 26 2013

nonn

The corresponding x-values are in A230622.

			a(2)=15 because 8^2 + 15^2 = 289 = 17^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, Oliver Riesen, On primitive solutions of the Diophantine equation x^2+ y^2= M, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).

Cf. A001026, A188949, A230622.

Table[Select[PowersRepresentations[17^n, 2, 2], CoprimeQ@@#&][[1, 2]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

1, 3, 2, 7, 38, 44, 29, 336, 718, 237, 2642, 10296, 8839, 16124, 108691, 164833, 24478, 922077, 2521451, 1476984, 6699319, 34182196, 35553398, 32125393, 306268562, 597551756, 130656229, 2465133864, 8701963882, 6890111163, 15949374758, 98248054847, 135250416961

Offset: 1

Colin Barker, Oct 28 2013

nonn

The corresponding y-values are in A230711.
For all non-coprime solutions (x,y) to the equation x^2 + y^2 = p^n, x and y are both divisible by the prime p.
Using de Moivre's Theorem (in essence), define (c,d)*(e,f) as (ce-df,cf+de). Then a(n) = min{|u(n)|, |v(n)|}, where (u(n),v(n)) = (2,1)^n = (2,1)*(2,1)^[n-1]. Proof: It can be readily seen that u^2(n) + v^2(n) = 5^n. To show that u(n) and v(n) are relatively prime, assume that x,y are relatively prime. Then (2,1)*(x,y) = (2x-y, x+2y). If a prime p were to divide both of 2x-y and x+2y, then p would divide 5y, so p=5. Now suppose x == 2 (mod 5) and y == 1 (mod 5). It can be seen that 2x-y == -2 (mod 5) and x+2y == -1 (mod 5). The reverse also holds. Because u(1)=2 and v(1)=1, the result follows inductively. - Richard Peterson, May 21 2021

			a(4)=7 because 7^2 + 24^2 = 625 = 5^4.

Zak Seidov, Table of n, a(n) for n = 1..200
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. A000351, A230711.
Cf. A188948, A230622, A230644.
Cf. A139011.

Table[Select[PowersRepresentations[5^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1,1]], {n, 33}] (* T. D. Noe, Nov 04 2013 *)

Typo in data fixed by Colin Barker, Nov 02 2013

A230644 Values of x such that x^2 + y^2 = 29^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

2, 20, 65, 41, 1475, 15939, 60422, 68880, 1063438, 12631900, 55535695, 86719879, 743336365, 9948240141, 50564635018, 96971881440, 496655601122, 7778740572980, 45669244934945, 101577438080201, 308633722311395, 6032082927439779, 40960925870541542

Offset: 1

Colin Barker, Oct 26 2013

nonn

The corresponding y-values are in A230645.

			a(3)=65 because 65^2+142^2=24389=29^3.

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

Cf. A009973, A188948, A230622, A230645.

Table[Select[PowersRepresentations[29^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

Incorrect formula deleted by Colin Barker, Jan 08 2014

A230712 Values of x such that x^2 + y^2 = 37^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

1, 12, 107, 840, 5646, 27755, 124158, 462961, 961686, 5589325, 102654282, 1025046359, 8502347874, 64101459205, 356029844147, 1681548425760, 7005476875681, 21848430755052, 2978524660427, 772649642011800

Offset: 1

Colin Barker, Oct 28 2013

nonn

The corresponding y-values are in A230713.

			a(3)=107 because 107^2+198^2=50653=37^3.

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

Cf. A009981, A188948, A230622, A230644, A230710, A230713.

Table[Select[PowersRepresentations[37^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

A230743 Values of x such that x^2 + y^2 = 41^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

4, 9, 115, 720, 2476, 42471, 4765, 1788961, 3780956, 51872200, 310486445, 1142532559, 18483128564, 4205436520, 799862636324, 1584162310079, 23384002313285, 133802323596440, 526151093402156, 8041209044472401, 2783579583540395, 357525366658772391

Offset: 1

Colin Barker, Oct 29 2013

nonn

The corresponding y-values are in A230744.

			a(3)=115 because 115^2+236^2=68921=41^3.

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

Cf. A009985, A230744, A188948, A230622, A230644, A230710, A230712.

Table[Select[PowersRepresentations[41^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

A230759 Values of x such that x^2 + y^2 = 53^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

2, 28, 259, 1241, 3647, 14715, 399301, 4810319, 34161842, 146769868, 244200526, 4359995640, 73982566838, 804676166812, 4381447604821, 15981352647839, 8477785985767, 965700694136205, 13070487060661219, 114948480102611400, 541029996598203398

Offset: 1

Colin Barker, Oct 29 2013

nonn

The corresponding y-values are in A230760.

			a(3)=259 because 259^2+286^2=148877=53^3.

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

Cf. A230760, A188948, A230622, A230644, A230710, A230712, A230743.

Table[Select[PowersRepresentations[53^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)

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

Original entry on oeis.org

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

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

Original entry on oeis.org

3, 48, 296, 721, 10072, 213785, 1958709, 7613760, 21165597, 894454032, 12278087704, 59926173839, 62518379032, 3374316625735, 58552907681096, 416603004343680, 1261259807092797, 10231862403603888, 255781764375436389, 2697529798981443601, 11543491568219853608

Offset: 1

Colin Barker, Nov 02 2013

nonn

The corresponding y-values are in A230963.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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. A230963, A188948, A230622, A230644, A230710, A230712, A230743, A230759, A230841.

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

Table[Select[PowersRepresentations[73^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
Table[Min[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([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

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

cp's OEIS Frontend

A230623 Values of y such that x^2 + y^2 = 17^n with x and y coprime and 0 < x < y.

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A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

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A230644 Values of x such that x^2 + y^2 = 29^n with x and y coprime and 0 < x < y.

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A230712 Values of x such that x^2 + y^2 = 37^n with x and y coprime and 0 < x < y.

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A230743 Values of x such that x^2 + y^2 = 41^n with x and y coprime and 0 < x < y.

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A230759 Values of x such that x^2 + y^2 = 53^n with x and y coprime and 0 < x < y.

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

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