A188982 -id:A188982 - cp's OEIS frontend

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

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

Zak Seidov, Apr 10 2011

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

			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.

Robert Israel, Table of n, a(n) for n = 0..3581

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.

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

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 *)

A188983 Odd numbers y such that x^2 + y^2 = 13^n with x and y coprime.

Original entry on oeis.org

1, 3, 5, 9, 119, 597, 2035, 4449, 239, 56403, 341525, 1315911, 3455641, 3627003, 23161315, 186118929, 815616479, 2474152797, 4241902555, 6712571031, 95420159401, 485257533003, 1671083125805, 3718150825791, 584824319281, 44827014819597, 276564805068235, 1076637637754649, 2864483360640839, 3190610873034597, 18094618450123325

Offset: 0

T. D. Noe, Apr 14 2011

nonn

The x values are in A188982.

This is also the absolute value of the real part of (3+2i)^n, where i = sqrt(-1). The signed version is A121622.

Vincenzo Librandi, Table of n, a(n) for n = 0..183

Cf. A121622, A158936, A188982.

[Integers()!Abs(Real((3+2*Sqrt(-1))^n)): n in [0..30]]; // Bruno Berselli, May 26 2011

Table[s = Select[PowersRepresentations[13^n, 2, 2], CoprimeQ @@ # &, 1][[1]]; Select[s, OddQ][[1]], {n, 0, 30}]

A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 0, -1, -2, -1, -1, -1, 2, 2, 3, 4, 3, 3, 3, 2, 1, 0, 1, 1, 1, 5, 5, 6, 7, 6, 6, 6, 5, 4, 3, 4, 4, 4, 8, 8, 9, 10, 9, 9, 9, 8, 7, 6, 7, 7, 7, 3, 3, 4, 5, 4, 4, 4, 3, 2, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 1, 0, -1, 0, 0, 0, -1, -1, 0, 1, 0, 0

Offset: 0

Rémy Sigrist, Oct 27 2021

The function f defines a bijection from the nonnegative integers to the Gaussian integers.
The following diagram depicts g(d) for d = 0..12:
                      |
                      |    +
                      |     3
                      |
            +    +    +    +
             6    5   |4    2
                      |
         --------+----+----+-------
                  7   |0    1
                      |
                 +    +    +    +
                  8   |10   11   12
                      |
                 +    |
                  9   |

Rémy Sigrist, Table of n, a(n) for n = 0..2197
Stephen K. Lucas, Base 2 + i with digit set {0, +/-1, +/-i}, ResearchGate (October 2021).
Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)

See A316658 for a similar sequence.

Cf. A188982, A348652.

g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
a(n) = imag(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))

abs(a(13^k)) = A188982(k).

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

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A188983 Odd numbers y such that x^2 + y^2 = 13^n with x and y coprime.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

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cp's OEIS Frontend

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

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A188983 Odd numbers y such that x^2 + y^2 = 13^n with x and y coprime.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.