cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A188948 Values of x such that x^2 + y^2 = 13^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

0, 2, 5, 9, 119, 122, 828, 4449, 239, 56403, 145668, 246046, 3369960, 3627003, 23161315, 128629846, 13651680, 1590277918, 4241902555, 6712571031, 95420159401, 107655263398, 647549275812, 3718150825791, 584824319281, 44827014819597, 123471611274972, 182714776311554, 2701419604443960, 3190610873034597, 18094618450123325
Offset: 0

Views

Author

Zak Seidov, Apr 10 2011

Keywords

Comments

The y values are in A188949.

Links

Crossrefs

Cf. A158936, A188949.

Programs

  • Mathematica
    Table[Select[PowersRepresentations[13^n, 2, 2], CoprimeQ @@ # &][[1,1]], {n, 0, 30}]

Extensions

Edited by T. D. Noe, Apr 14 2011

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

Original entry on oeis.org

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

Views

Author

Colin Barker, Oct 26 2013

Keywords

Comments

The corresponding x-values are in A230622.

Examples

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

Links

Crossrefs

Cf. A001026, A188949, A230622.

Programs

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

A230645 Values of y such that x^2 + y^2 = 29^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

5, 21, 142, 840, 4282, 18460, 116615, 703919, 3657355, 16159899, 95479298, 588467880, 3115779158, 14092223060, 77925646825, 490757504161, 2647731283685, 12245345216181, 63384393297262, 408260456356200, 2244457157941402, 10605018345084220
Offset: 1

Views

Author

Colin Barker, Oct 26 2013

Keywords

Comments

The corresponding x-values are in A230644.

Examples

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

Crossrefs

Cf. A009973, A188949, A230623, A230644.

Extensions

Incorrect formula deleted by Colin Barker, Jan 08 2014

A230711 Values of y such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

2, 4, 11, 24, 41, 117, 278, 527, 1199, 3116, 6469, 11753, 33802, 76443, 136762, 354144, 873121, 1721764, 3565918, 9653287, 20783558, 34867797, 103232189, 242017776, 451910159, 1064447283, 2726446322, 5583548873, 10513816601, 29729597084, 66349305331
Offset: 1

Views

Author

Colin Barker, Oct 28 2013

Keywords

Comments

The corresponding x-values are in A230710.

Examples

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

Links

Crossrefs

Cf. A000351, A188949, A230623, A230645, A230710.

Programs

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

A158936 List of coprime pairs (x,y) such that x^2+y^2 = 13^n with 0

Original entry on oeis.org

0, 1, 2, 3, 5, 12, 9, 46, 119, 120, 122, 597, 828, 2035, 4449, 6554, 239, 28560, 56403, 86158, 145668, 341525, 246046, 1315911, 3369960, 3455641, 3627003, 17021162, 23161315, 58317492, 128629846, 186118929, 13651680, 815616479, 1590277918, 2474152797, 4241902555, 9719139348, 6712571031, 37641223154, 95420159401, 99498527400, 107655263398, 485257533003
Offset: 0

Views

Author

Zak Seidov, Apr 10 2011

Keywords

Comments

For n>2, all other solutions (x,y) are divisible by 13, e.g., 26^2+39^2=13^3.

Examples

			n=1: 13^1=2^2+3^2, hence a(1)=2, a(2)=3,
n=2: 13^2=5^2+12^2, hence a(3)=5, a(4)=12.

Links

Crossrefs

Cf. A098122 for case x^2+y^2=5^n.
Cf. A188948, A188949 for the values of x and y separately.
Cf. A188982, A188983 for even and odd terms.

Programs

  • Maple
    f:= proc(n) local q;
  q:= map(abs, [Re,Im]((2+3*I)^n));
  op(sort(q))
end proc:
map(f, [$0..50]); # Robert Israel, Feb 27 2024
  • Mathematica
    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
Table[Select[PowersRepresentations[13^n, 2, 2], CoprimeQ @@ # &][[1]], {n, 0, 21}] (* T. D. Noe, Apr 12 2011 *)

Formula

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

A230713 Values of y such that x^2 + y^2 = 37^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

6, 35, 198, 1081, 6121, 42372, 281987, 1816080, 11359441, 69118332, 409120667, 2352069720, 13087371961, 70021883892, 454630639122, 3083813678879, 20184430499034, 128112059869885, 790520789974362, 4746103264506599
Offset: 1

Views

Author

Colin Barker, Oct 28 2013

Keywords

Comments

The corresponding x-values are in A230712.

Examples

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

Crossrefs

Cf. A009981, A188949, A230623, A230645, A230711, A230712.

A230744 Values of y such that x^2 + y^2 = 41^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

5, 40, 236, 1519, 10475, 54280, 441284, 2187360, 17694245, 103595049, 673741196, 4610651760, 24155269835, 194708863431, 956722571075, 7826203465920, 45467666569916, 298790677846089, 2029162683616205, 10747406201475600, 85901867185267604, 418375017592176440
Offset: 1

Views

Author

Colin Barker, Oct 29 2013

Keywords

Comments

The corresponding x-values are in A230743.

Examples

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

Crossrefs

Cf. A009985, A230743.
Cf. A188949, A230623, A230645, A230711, A230713.

A230760 Values of y such that x^2 + y^2 = 53^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

7, 45, 286, 2520, 20122, 148148, 1007606, 6254640, 46181513, 391594275, 3034699661, 21731298679, 143399099473, 855828562635, 7344390292954, 60173627260320, 453178096117918, 3155291100853892, 20155636317704834, 131804682060038201, 1152529734625490207
Offset: 1

Views

Author

Colin Barker, Oct 29 2013

Keywords

Comments

The corresponding x-values are in A230759.

Examples

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

Crossrefs

Cf. A230759.
Cf. A188949, A230623, A230645, A230711, A230713, A230744.

A230842 Values of y such that x^2 + y^2 = 61^n with x and y coprime and 0 < x < y.

Original entry on oeis.org

6, 60, 415, 3479, 27474, 194220, 1752665, 10361041, 108089046, 665045051, 6449019055, 45629658360, 371682946434, 3000926368429, 20605153668425, 190323205453920, 1089586196530086, 11672337637546091, 73603293662217846, 692487869011494600, 5011061854350480605
Offset: 1

Views

Author

Colin Barker, Oct 31 2013

Keywords

Comments

The corresponding x-values are in A230841.

Examples

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

Crossrefs

Cf. A230841.
Cf. A188949, A230623, A230645, A230711, A230713, A230744, A230760.

Programs

  • Mathematica
    Table[Select[PowersRepresentations[61^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1, 2]], {n, 21}] (* T. D. Noe, Nov 04 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

Views

Author

Colin Barker, Nov 02 2013

Keywords

Comments

The corresponding x-values are in A230962.

Examples

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

Links

Crossrefs

Cf. A230962.
Cf. A188949, A230623, A230645, A230711, A230713, A230744, A230760, A230842.

Programs

  • Maple
    f:=n ->  max([abs@Re,abs@Im]((3+8*I)^n)):
map(f, [$1..50]); # Robert Israel, Mar 31 2017
  • Mathematica
    Table[Max[Abs[Re[(3 + 8I)^n]], Abs[Im[(3 + 8I)^n]]], {n, 30}] (* Indranil Ghosh, Mar 31 2017, after formula by Robert Israel *)
  • Python
    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

Formula

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)
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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