A243188 -id:A243188 - cp's OEIS frontend

A107152 Primes of the form x^2 + 45y^2.

61, 109, 181, 229, 241, 349, 409, 421, 541, 601, 661, 709, 769, 829, 1009, 1021, 1069, 1129, 1201, 1249, 1321, 1381, 1429, 1489, 1549, 1609, 1621, 1669, 1741, 1789, 1801, 1861, 2029, 2089, 2161, 2221, 2269, 2281, 2341, 2389, 2521, 2689, 2749, 3001, 3049, 3061, 3109, 3121, 3169, 3181

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -180. See A107132 for more information.
Also primes of the form x^2 + 60y^2. See A140633. - T. D. Noe, May 19 2008
Also primes of the form x^2+6*x*y-6*y^2, of discriminant 60 (as well as of the form x^2+8*x*y+y^2). -  Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A139643.
Cf. A141302, A141303, A141304 (d=60).
All representatives in A243188.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(3000) | p mod 60 in {1, 49 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[1, 0, 45, 10000] (* see A106856 *)
Select[Prime[Range[500]], MatchQ[Mod[#, 60], 1|49]&] (* Jean-François Alcover, Oct 28 2016 *)

list(lim)=my(v=List(),t); forprime(p=61,lim, t=p%60; if(t==1||t==49, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

Primes congruent to {1, 49} (mod 60). - T. D. Noe, Apr 29 2008

A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

Original entry on oeis.org

1, 4, 5, 9, 11, 16, 19, 20, 25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76, 79, 80, 81, 89, 95, 99, 100, 101, 109, 116, 121, 124, 125, 131, 139, 144, 145, 149, 151, 155, 164, 169, 171, 176, 179, 180, 181, 191, 196, 199, 205, 209, 211, 220, 225, 229, 236

Offset: 1

N. J. A. Sloane

5x^2 - y^2 has discriminant 20, x^2 + xy - y^2 has discriminant 5. - N. J. A. Sloane, May 30 2014
Representable as x^2 + 3xy + y^2 with 0 <= x <= y. - Benoit Cloitre, Nov 16 2003
Numbers k such that x^2 - 3xy + y^2 + k = 0 has integer solutions. - Colin Barker, Feb 04 2014
Numbers k such that x^2 - 7xy + y^2 + 9k = 0 has integer solutions. - Colin Barker, Feb 10 2014
Also positive numbers of the form x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

M. Baake, "Solution of coincidence problem ...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Robert Israel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG], 2006.
M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), this sequence, A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), A316621(k=9).
See A035187 for number of representations.
Primes in this sequence: A038872, also A141158.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
See also the related sequence A263849 based on a theorem of Maass.

select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]); # Robert Israel, Jun 12 2014

ok[n_] := Resolve[Exists[{x, y}, Element[x|y, Integers], n == 5*x^2-y^2]]; Select[Range[236], ok]
(* or, for a large number of terms: *)
max = 60755 (* max=60755 yields 10000 terms *); A031363 = {}; xm = 1;
While[T = A031363; A031363 = Table[5*x^2 - y^2, {x, 1, xm}, {y, 0, Floor[ x*Sqrt[5]]}] // Flatten // Union // Select[#, # <= max&]&; A031363 != T, xm = 2*xm]; A031363  (* Jean-François Alcover, Mar 21 2011, updated Mar 17 2018 *)

select(x -> x, direuler(p=2,101,1/(1-(kronecker(5,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020, after hints by Colin Barker, Jun 18 2014, and Michel Marcus

is(n)=#bnfisintnorm(bnfinit(z^2-z-1),n) \\ Ralf Stephan, Oct 18 2013

seq(M,k=3) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
seq(236) \\ Gheorghe Coserea, Jul 29 2018

from itertools import count, islice
from sympy import factorint
def A031363_gen(): # generator of terms
    return filter(lambda n:all(not((1 < p % 5 < 4) and e & 1) for p, e in factorint(n).items()),count(1))
A031363_list = list(islice(A031363_gen(),30)) # Chai Wah Wu, Jun 28 2022

Consists exactly of numbers in which primes == 2 or 3 mod 5 occur with even exponents.

Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = 5.

More terms from Erich Friedman

b-file corrected and extended by Robert Israel, Jun 12 2014

A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

6, 11, 14, 15, 24, 35, 44, 51, 54, 56, 59, 60, 71, 86, 96, 99, 110, 119, 126, 131, 134, 135, 140, 150, 159, 176, 179, 191, 204, 206, 215, 216, 224, 231, 236, 239, 240, 251, 254, 275, 284, 294, 311, 315, 326, 335, 339, 344, 350, 359, 366, 371, 374, 375, 384

Offset: 1

Colin Barker, Feb 10 2014

nonn

From Klaus Purath, Feb 17 2024: (Start)
Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n].
Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) =  k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.
Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.
Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.
Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)

			6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).

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

Cf. A070997 (k = 6), A199336 (k = 14), A001091 (k = 15), A077248 (k = 35).
Cf. A031363, A084917, A237351, A237599, A237609, A237610.
For primes see A141302.
Cf. A378710, A378711 (subsequence of properly represented numbers and fundamental solutions).

A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.

Original entry on oeis.org

0, 2, 5, 8, 17, 18, 20, 32, 33, 42, 45, 50, 53, 68, 72, 77, 80, 98, 105, 113, 122, 125, 128, 132, 137, 153, 162, 168, 170, 173, 177, 180, 197, 200, 212, 213, 218, 233, 242, 245, 257, 258, 272, 288, 293, 297, 305, 308, 317, 320, 330, 338, 353, 357, 362, 378

Offset: 1

N. J. A. Sloane, Jun 05 2014

nonn

Discriminant 60.
Nonnegative numbers of the form 5x^2 - 3y^2. - Jon E. Schoenfield, Jun 03 2022
From Klaus Purath, Jul 26 2023: (Start)
Nonnegative integers k such that 3x^2 - 5y^2 + k = 0 has integer solutions.
Also nonnegative integers of the form 2x^2 + (4m+2)xy + (2m^2+2m-7)y^2 for integers m. This includes the form in the name with m = 1.
Also nonnegative integers of the form 5x^2 + 10mxy + (5m^2-3)y^2 for integers m. This includes the form from Jon E. Schoenfield above with m = 0.
There are no squares in this sequence. Even powers of terms as well as products of an even number of terms belong to A243188.
Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the form 5x^2 - 3y^2 by the following identity: (na^2 - kb^2)(nc^2 - kd^2)(ne^2 - kf^2) = n[a(nce + kdf) + bk(cf + de)]^2 - k[na(cf + de) + b(nce + kdf)]^2 for all a, b, c, d, e, f, k, n in R. This can be verified by expanding both sides of the equation.
(End)

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

Primes: A141303. Cf. A243188, A107152, A237606, A141302, A243190, A141304.

Reap[For[n = 0, n <= 200, n++, If[Reduce[2*x^2 + 6*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

0 prepended and more terms from Colin Barker, Apr 07 2015

A243190 Nonnegative numbers of the form -2x^2+6xy+3y^2.

Original entry on oeis.org

0, 3, 7, 12, 22, 27, 28, 30, 43, 48, 55, 63, 67, 70, 75, 88, 102, 103, 108, 112, 118, 120, 127, 142, 147, 163, 172, 175, 183, 187, 192, 198, 220, 223, 238, 243, 252, 255, 262, 268, 270, 280, 283, 295, 300, 307, 318, 327, 343, 352, 355, 358, 363, 367, 382

Offset: 1

N. J. A. Sloane, Jun 05 2014

nonn

Discriminant 60.

Also: nonnegative 3x^2-5y^2 since 3y^2+6xy-2x^2 = 3(y+x)^2-5x^2. - R. J. Mathar, Jun 10 2020

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

Primes: A141304. Cf. A243188, A107152, A237606, A141302, A243189, A141303.

Reap[For[n = 0, n <= 200, n++, If[Reduce[-2*x^2 + 6*x*y + 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

0 prepended and more terms from Colin Barker, Apr 07 2015

A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.

Original entry on oeis.org

0, 1, 4, 9, 11, 16, 23, 25, 36, 37, 44, 49, 53, 64, 67, 71, 81, 91, 92, 99, 100, 113, 119, 121, 133, 137, 144, 148, 163, 169, 176, 179, 191, 196, 207, 212, 221, 225, 247, 253, 256, 268, 275, 284, 287, 289, 317, 323, 324, 331, 333, 361, 364, 368, 379, 389, 396, 400, 401, 407, 421, 427, 441, 443, 449

Offset: 1

Gheorghe Coserea, Jul 29 2018

nonn

Discriminant 77.

In general, for k>=0 the positive part of the set S = {x^2 - k*x*y + y^2: x,y in Z} is given by the numbers of the form x^2 + k*x*y + y^2 with 0 <= x <= y natural numbers.

Gheorghe Coserea, Table of n, a(n) for n = 1..100001

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), A031363(k=3), A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), this sequence.

seq(M,k=9) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
concat(0, seq(449))

cp's OEIS Frontend

A107152 Primes of the form x^2 + 45y^2.

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A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

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A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

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A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.

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A243190 Nonnegative numbers of the form -2x^2+6xy+3y^2.

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A316621 Numbers of the form x^2 + 9*x*y + y^2, 0 <= x <= y.

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A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.