A033212 -id:A033212 - cp's OEIS frontend

A141785 Primes of the form -x^2 + 5xy + 5y^2 (as well as of the form 9x^2 + 15xy + 5*y^2).

5, 11, 29, 41, 59, 71, 89, 101, 131, 149, 179, 191, 239, 251, 269, 281, 311, 359, 389, 401, 419, 431, 449, 461, 479, 491, 509, 521, 569, 599, 641, 659, 701, 719, 761, 809, 821, 839, 881, 911, 929, 941, 971, 1019, 1031, 1049, 1061, 1091, 1109, 1151, 1181, 1229, 1259

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 45. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(2) = 29 because we can write 29 = -1^2 + 5*1*2 + 5*2^2 (or 29 = 9*1^2 + 15*1*1 + 5*1^2)

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

Juan Arias-de-Reyna, 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. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A033212 (d=45), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 5 || MatchQ[Mod[#, 45], Alternatives[11, 14, 26, 29, 41, 44]]&] (* Jean-François Alcover, Oct 28 2016 *)

A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Reduced form is [1, 3, -3]. Discriminant = 21. Class number = 2.
Values of the quadratic form are {0, 1, 3, 4} mod 6, so this is a subsequence of A002476. - R. J. Mathar, Jul 30 2008
It can be checked that the primes p of the form x^2 + n*x*y + y^2, n >= 3, where x and y are nonnegative, depend on n mod 6 as follows: n mod 6 = 0 => p mod 12 = {1,5}; n mod 6 = 1 => p mod 12 = {1,7}; n mod 6 = 2 => p mod 12 = {1}; n mod 6 = 3 => p mod 12 = {1,5,7,11}; n mod 6 = 4 => p mod 12 = {1}; n mod 6 = 5 => p mod 12 = {1,7}. - Walter Kehowski, Jun 01 2008

			a(1) = 7 because we can write 7 = 1^2 + 5*1*1 + 1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Peter Luschny, Binary Quadratic Forms
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.

Primes in A243172.
Cf. A002476, A007645, A007519, A033212, A033215, A068228, A107008, A107145, A107152, A139490, A139489, A139491, A139492, A141159.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a = {}; w = 5; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 5, 1])
print(Q.represented_positives(1303, 'prime')) # Peter Luschny, May 12 2021

A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

241, 409, 601, 769, 1009, 1129, 1201, 1249, 1321, 1489, 1609, 1801, 2089, 2161, 2281, 2521, 2689, 3001, 3049, 3121, 3169, 3361, 3529, 3769, 3889, 4129, 4201, 4441, 4561, 4729, 4801, 4969, 5209, 5281, 5449, 5521, 5569, 5641, 5689, 5881, 6121, 6361, 6481

Offset: 1

Artur Jasinski, Apr 24 2008

Also primes of the form x^2 + 120y^2. - T. D. Noe, Apr 29 2008
Also primes of the form x^2+240y^2. See A140633. - T. D. Noe, May 19 2008
In base 12, the sequence is 181, 2X1, 421, 541, 701, 7X1, 841, 881, 921, X41, E21, 1061, 1261, 1301, 13X1, 1561, 1681, 18X1, 1921, 1981, 1X01, 1E41, 2061, 2221, 2301, 2481, 2521, 26X1, 2781, 28X1, 2941, 2X61, 3021, 3081, 31X1, 3241, 3281, 3321, 3361, 34X1, 3661, 3821, 3901, where X is 10 and E is 11. Moreover, the discriminant is 340. - Walter Kehowski, Jun 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

[ p: p in PrimesUpTo(7000) | p mod 120 in {1, 49}]; // Vincenzo Librandi, Jul 28 2012

QuadPrimes2[1, 0, 120, 10000] (* see A106856 *)

The primes are congruent to {1, 49} (mod 120). - T. D. Noe, Apr 29 2008

A243173 Numbers of the form x^2+15y^2.

Original entry on oeis.org

0, 1, 4, 9, 15, 16, 19, 24, 25, 31, 36, 40, 49, 51, 60, 61, 64, 69, 76, 79, 81, 85, 96, 100, 109, 115, 121, 124, 135, 136, 139, 141, 144, 151, 159, 160, 169, 171, 181, 184, 196, 199, 204, 211, 216, 225, 229, 235, 240, 241, 244, 249, 256, 265, 271, 276, 279, 285, 289, 304, 316, 321, 324, 331, 339, 340, 349, 360, 361, 375, 376, 379, 384, 391, 400, 409, 411

Offset: 1

N. J. A. Sloane, Jun 01 2014

nonn

Discriminant -60.

Norms of numbers in Z[sqrt(-15)]. - Alonso del Arte, Sep 23 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes: A033212.

nn = 22; Union[Select[Flatten[Table[x^2 + 15 y^2, {x, 0, nn}, {y, 0, nn}]], # <= nn^2 &]] (* Bruno Berselli, Jun 02 2014 *)

A243174 Nonnegative integers of the form x^2 + 5xy - 5*y^2 (discriminant 45).

Original entry on oeis.org

0, 1, 4, 9, 16, 19, 25, 31, 36, 45, 49, 55, 61, 64, 76, 79, 81, 99, 100, 109, 121, 124, 139, 144, 145, 151, 169, 171, 180, 181, 196, 199, 205, 211, 220, 225, 229, 241, 244, 256, 261, 271, 279, 289, 295, 304, 316, 319, 324, 331, 349, 355, 361, 369, 379, 396, 400, 405, 409, 421, 436, 439, 441, 445, 451, 475, 484, 495, 496, 499, 505, 529, 531, 541, 549

Offset: 1

N. J. A. Sloane, Jun 01 2014

nonn

Also numbers representable as x^2 + 7*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - x*y - 11*y^2 with 0 <= x <= y (or x^2 + x*y - 11*y^2 with x, y nonnegative). - Jianing Song, Jul 31 2018
Also nonnegative integers of the form 9x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

Gheorghe Coserea, Table of n, a(n) for n = 1..100001
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A028957, A031363.

Primes: A033212.

A028957 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 8 ] (divided by 2).

Original entry on oeis.org

0, 1, 4, 6, 9, 10, 15, 16, 19, 24, 25, 31, 34, 36, 40, 46, 49, 51, 54, 60, 61, 64, 69, 76, 79, 81, 85, 90, 94, 96, 100, 106, 109, 114, 115, 121, 124, 135, 136, 139, 141, 144, 150, 151, 159, 160, 166, 169, 171, 181, 184, 186, 190, 196, 199, 204, 211, 214, 216, 225

Offset: 1

N. J. A. Sloane

nonn

Nonnegative integers of the form x^2 + x*y + 4y^2, discriminant -15. - Ray Chandler, Jul 12 2014

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

Primes are in A033212.

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

13, 43, 61, 79, 103, 127, 139, 157, 181, 199, 211, 277, 283, 313, 337, 367, 373, 433, 439, 523, 547, 571, 601, 607, 673, 727, 751, 757, 823, 829, 859, 883, 907, 919, 937, 991, 997, 1039, 1063, 1069, 1093, 1117, 1153, 1171, 1213, 1231, 1249, 1291, 1297

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

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

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 11; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

193, 337, 457, 673, 1009, 1033, 1129, 1201, 1297, 1801, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2713, 2857, 3049, 3217, 3313, 3361, 3529, 3697, 3889, 4057, 4153, 4201, 4561, 4657, 4729, 4993, 5209, 5233, 5569, 5737, 5881, 6073, 6217, 6337, 6553, 6577

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Also primes of the form x^2 + 168y^2. - T. D. Noe, Apr 29 2008

In base 12, the sequence is 141, 241, 321, 481, 701, 721, 7X1, 841, 901, 1061, 1101, 1201, 12X1, 1461, 1521, 1561, 1681, 16X1, 17X1, 1921, 1X41, 1E01, 1E41, 2061, 2181, 2301, 2421, 24X1, 2521, 2781, 2841, 28X1, 2X81, 3021, 3041, 3281, 33X1, 34X1, 3621, 3721, 3801, 3961, 3981, where X is 10 and E is 11. Moreover, the discriminant is 480. - Walter Kehowski, Jun 01 2008

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

Cf. A139489, A007645, A068228, A007519, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

a = {}; w = 26; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

The primes are congruent to {1, 25, 121} (mod 168). - T. D. Noe, Apr 29 2008

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.

Original entry on oeis.org

229, 349, 409, 421, 661, 769, 829, 1021, 1069, 1249, 1381, 1429, 1549, 1789, 1801, 1861, 2089, 2161, 2269, 2389, 3001, 3061, 3109, 3181, 3229, 3469, 3889, 4021, 4129, 4201, 4441, 4861, 4909, 5101, 5449, 5521, 5869, 5881, 6121, 6469, 6481, 6529, 6781

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Are all terms == 1 mod 12? - Zak Seidov, Apr 25 2008

Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - R. J. Mathar, Jun 10 2020

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references), discriminant 1020.

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A106858 Primes of the form 2x^2+xy+2y^2 with x and y nonnegative.

Original entry on oeis.org

2, 5, 23, 83, 107, 137, 173, 257, 293, 347, 353, 467, 503, 617, 647, 653, 743, 797, 857, 953, 983, 1223, 1277, 1283, 1307, 1427, 1487, 1493, 1523, 1553, 1637, 1787, 1877, 1913, 1997, 2003, 2027, 2213, 2237, 2243, 2393, 2423, 2447, 2657, 2663

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-15.

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 1000 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A028955, A106859, A243173, A033212.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
t2 = QuadPrimes2[2, 1, 2, 350000];
Length[t2]
t2[[Length[t2]]]
For[n=1, n <= 2000, n++, Print[n, " ", t2[[n]]]] (* From N. J. A. Sloane, Jun 17 2014 *)

Replace Mma program by a correct program, recomputed and extended b-file. - N. J. A. Sloane, Jun 17 2014

cp's OEIS Frontend

A141785 Primes of the form -x^2 + 5*x*y + 5*y^2 (as well as of the form 9*x^2 + 15*x*y + 5*y^2).

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A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

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A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

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Magma

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A243173 Numbers of the form x^2+15y^2.

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A243174 Nonnegative integers of the form x^2 + 5*x*y - 5*y^2 (discriminant 45).

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A028957 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 8 ] (divided by 2).

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Extensions

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

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Mathematica

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

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Mathematica

Formula

A139512 Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

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A141785 Primes of the form -x^2 + 5xy + 5y^2 (as well as of the form 9x^2 + 15xy + 5*y^2).

A243174 Nonnegative integers of the form x^2 + 5xy - 5*y^2 (discriminant 45).

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.