A244819 -id:A244819 - cp's OEIS frontend

A244780 Positive numbers primitively represented by the binary quadratic form (1, 1, 3).

1, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 37, 45, 47, 53, 55, 59, 67, 69, 71, 75, 81, 89, 93, 97, 99, 103, 111, 113, 115, 125, 135, 137, 141, 155, 157, 159, 163, 165, 177, 179, 181, 185, 191, 199, 201, 207, 213, 223, 225, 229, 235, 243, 251, 253, 257, 265, 267

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -11.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A244779, A244819.

# Function PriRepBQF in A244779.
A244780_list := n -> PriRepBQF(1, 1, 3, n); A244780_list(267);

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y + 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A244779 Positive numbers primitively represented by the binary quadratic form (1, 1, 2).

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 14, 16, 22, 23, 28, 29, 32, 37, 43, 44, 46, 53, 56, 58, 64, 67, 71, 74, 77, 79, 86, 88, 92, 106, 107, 109, 112, 113, 116, 121, 127, 128, 134, 137, 142, 148, 149, 151, 154, 158, 161, 163, 172, 176, 179, 184, 191, 193, 197, 203, 211, 212, 214

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -7.

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A244780, A244819.

PriRepBQF := proc(a, b, c, n) local L,q,R,r,k;
q := a*x^2 + b*x*y + c*y^2; L := NULL;
for k from 1 to n do
   R := [isolve(q = k)];
   if R = [] then next fi;
   for r in R do
      igcd(op(2,r[1]), op(2,r[2]));
      if 1 = % then L := L,k; break fi od
od; L end:
A244779_list := n -> PriRepBQF(1, 1, 2, n); A244779_list(214);
# Alternate program
A244779_set:= proc(N) local A, B, y,x;
   A:= {};
   for y from 0 to floor(sqrt(4*N/7)) do
     for x from ceil(-y/2) to floor(-y/2 + sqrt(N - 7/4*y^2)) do
       if igcd(x,y) = 1 then
         A:= A union {x^2 + x*y + 2*y^2}
       fi
     od
    od;
A
end proc:
A244779_set(1000); # Robert Israel, Jul 06 2014

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y + 2 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

Original entry on oeis.org

4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259, 268

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Superset of primes of the form 6n+1 (A002476).

For all proper solutions with nonnegative x and y see A244819. - Wolfdieter Lang, Mar 02 2021

E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092572, A092573, A092575, A244819.

A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

Original entry on oeis.org

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 39, 43, 52, 57, 61, 67, 73, 76, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 151, 157, 163, 172, 181, 183, 193, 199, 201, 211, 217, 228, 237, 241, 244, 247, 259, 268, 271, 273, 277, 283, 292, 301, 307, 309, 313, 327, 331, 337, 343, 364

Offset: 1

Robin Visser, Nov 02 2024

nonn

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

			The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [1, 2]], [3, [0, 1]], [4, [1, 1]], [7, [2, 1]], [12, [3, 1]], [13, [7, 2]], [19, [4, 1]], [21, [9, 2]], [28, [5, 1]], [31, [11, 2]], [39, [6, 1]], [43, [13, 2]], [52, [7, 1]], [57, [15, 2]], [61, [5639, 722]], [67, [8, 1]], [73, [17, 2]], [76, [61, 7]], [84, [9, 1]], [91, [19, 2]], [93, [135, 14]], [97, [847, 86]], [103, [10, 1]], [109, [1399, 134]], [111, [21, 2]], [124, [11, 1]], [127, [293, 26]], [129, [159, 14]], [133, [23, 2]], [139, [224, 19]], [147, [12, 1]], [151, [86, 7]], [157, [25, 2]], [163, [932, 73]], [172, [13, 1]], [181, [11262809, 837158]], [183, [27, 2]], [193, [189743, 13658]], [199, [14, 1]], ...

Robin Visser, Table of n, a(n) for n = 1..10000
Dario Alpern, Generic two integer variable equation solver.
Eric Weisstein's World of Mathematics, Pell Equation.

Cf. A031396, A244819, A261246, A377607.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377600_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d,-3)), count(max(startvalue,1)))
A377600_list = list(islice(A377600_gen(),63)) # Chai Wah Wu, Nov 03 2024

A341420 The positive integer numbers k represented properly by the binary quadratic form x^2 + 4*y^2.

Original entry on oeis.org

1, 4, 5, 8, 13, 17, 20, 25, 29, 37, 40, 41, 52, 53, 61, 65, 68, 73, 85, 89, 97, 100, 101, 104, 109, 113, 116, 125, 136, 137, 145, 148, 149, 157, 164, 169, 173, 181, 185, 193, 197, 200, 205, 212, 221, 229, 232, 233, 241, 244, 257, 260, 265, 269, 277, 281, 289, 292, 293, 296

Offset: 1

Wolfdieter Lang, Mar 19 2021

nonn

If also improper solutions of the Diophantine equation X^2 + Y^2 = k, with positive integer number k are taken into account one can obtain the present solutions provided X or Y are even. E.g., k = 4 has only improper solutions like (X, Y) = (0, pm2) or (pm2, 0) (pm stands for +1 or -1). So 4 is not a member of A008784, but in the present sequence it appears from (x, y) = (0, pm1) obtained from the first (X, Y) solution by y = Y/2.
The number k = 2 = A008784(2) is not represented here because there is only the proper solution (X, Y) = (pm1, pm1).
The number of solutions m(k = a(n)), up to an overall sign change in x and y, is given by m(1) = 1, m(4) = 1, m(8) = 2 and for k = 4^a*8^b*Product_{j=1..P1} (p1_j)^e1_j, with (a,b) from {(0, 0), (1, 0), (0, 1)}}, primes p1_j congruent to 1 (mod 4) (from A002144) and nonnegative exponents e1_j, it is m(k) = 2^(b + P1).
The primitive parallel binary quadratic forms of discriminant -16 = -4*4 representing positive integer numbers k are obtained by solving the Diophantine equation j^2 + 4 == 0 (mod k), for j from {0, 1, ..., k-1}. This gives for k = 1, 2, 4, and 8 the solutions j = 0, 0, {0, 2}, and {2, 6}, respectively. No larger powers of 2 have solutions. No lifting is possible (see Apostol, Theorem 5.30). For odd primes k the Legendre symbol (-4, k) = +1 exactly for k = prime == 1 (mod 4) (from the Legendre symbol (-1, prime) = +1 only for these primes A008784).
These parallel forms are given by (k, 2*j(k), c(j(k))), with c(j(k)) = (j(k)^2 + 4)/k.
There is only one primitive reduced form for discriminant -16, namely the principal form (1, 0, 4) (see the Buell reference p. 20). Thus each parallel form is equivalent (with a determinant +1 transformation) to this principal form, and gives a proper solution.

			Proper solutions (x, y) (up to overall sign flip) for various k = a(n):
a(2) = 4: (1, 0), m(4) = 1 (a = 1, b = 0, P1 = 0), (2, 0) is not a proper solution);
a(4) = 8: (2, pm1): (pm stands for +1 or -1), m(8) = 2 (a = 0, b = 1, P1 = 0);
a(7) = 20 = 4*5: (4, pm1), m(20) = 2 (a = 1, b = 0, P1 = 1), (m(4) = 1);
a(8) = 25 = 5^2: (3, pm2), m(25) = 2 (a = 0, b = 0, P1 = 1);
a(42) = 200 = 8*5^2: (2, pm7), (14, pm1), m(200) = 4 (a = 0, b = 1, P1 = 1).

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 120-122.
D. A. Buell, Binary Quadratic Forms, Springer, 1989, p. 20.

Cf. Discriminants -4: A008784, -8 A057127, -12 A244819.

a(n) = x(n)^2 + (2*y(n))^2, with gcd(x(n), y(n)) = 1, for n >= 1.

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A244780 Positive numbers primitively represented by the binary quadratic form (1, 1, 3).

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A244779 Positive numbers primitively represented by the binary quadratic form (1, 1, 2).

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Maple

Mathematica

A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

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A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

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Python

A341420 The positive integer numbers k represented properly by the binary quadratic form x^2 + 4*y^2.

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