A106856 -id:A106856 - cp's OEIS frontend

A033211 Primes of form x^2 + 14*y^2.

23, 127, 137, 151, 233, 239, 281, 359, 431, 449, 487, 673, 743, 751, 911, 953, 967, 977, 1033, 1087, 1103, 1129, 1303, 1409, 1423, 1439, 1481, 1663, 1759, 1871, 1873, 2017, 2039, 2081, 2129, 2137, 2207

Offset: 1

N. J. A. Sloane

Cohn, Harvey. A classical invitation to algebraic numbers and class fields. With two appendices by Olga Taussky: "Artin's 1932 Göttingen lectures on class field theory" and "Connections between algebraic number theory and integral matrices". Universitext. Springer-Verlag, New York-Heidelberg, 1978. xiii+328 pp. ISBN: 0-387-90345-3; MR0506156 (80c:12001). See p. 158.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989. See pp. ix, 115, etc.

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

Subsequence of A033207. Primes in A244037.

QuadPrimes2[1, 0, 14, 10000] (* see A106856 *)

A033235 Primes of the form x^2 + 55*y^2.

Original entry on oeis.org

59, 71, 199, 229, 251, 269, 311, 379, 389, 499, 509, 631, 661, 691, 751, 839, 881, 929, 1049, 1061, 1171, 1181, 1279, 1321, 1409, 1439, 1499, 1571, 1609, 1699, 1721, 1741, 1901, 1951, 2029, 2069, 2269

Offset: 1

N. J. A. Sloane

Also primes of the form x^2 - xy + 14y^2 with x and y nonnegative. - T. D. Noe, May 08 2005
From Lechoslaw Ratajczak, Apr 09 2017: (Start)
Conjecture: consecutive elements of this sequence are consecutive primes satisfying the congruence b(k) == 1 (mod k) for k>0, where b(k) is recursive sequence defined as follows: b(k) = -b(k-1) - b(k-2) + b(k-3) - b(k-4) with b(0)=2, b(1)=1, b(2)=0, b(3)=-1.
(b(59) - 1) mod 59 = (-496870918 - 1) mod 59 = 0, 59 = a(1).
(b(71) - 1) mod 71 = (88081764473 - 1) mod 71 = 0, 71 = a(2).
For 10^6 consecutive positive integers there are 9748 prime solutions and 5 nonprime (1, 586, 2935, 17161, 429737) solutions of the congruence. (End)

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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

QuadPrimes2[1, 0, 55, 10000] (* see A106856 *)

A033246 Primes of form x^2+71*y^2.

Original entry on oeis.org

71, 107, 293, 509, 643, 647, 739, 971, 1013, 1039, 1123, 1217, 1361, 1367, 1373, 1423, 1663, 1811, 2083, 2099, 2239, 2309, 2351, 2557, 2657, 2677, 2917, 3181, 3343, 3517, 3533, 3539, 3623, 3671, 3803

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+18y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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

QuadPrimes2[1, 0, 71, 10000] (* see A106856 *)

A033251 Primes of form x^2+79*y^2.

Original entry on oeis.org

79, 83, 179, 223, 317, 397, 479, 541, 563, 677, 727, 757, 811, 863, 907, 941, 967, 1103, 1277, 1289, 1433, 1489, 1523, 1553, 1867, 1889, 1979, 1993, 1997, 2011, 2039, 2311, 2341, 2383, 2459, 2551, 2579

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+20y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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

QuadPrimes2[1, 0, 79, 10000] (* see A106856 *)

A033256 Primes of form x^2+87*y^2.

Original entry on oeis.org

103, 151, 283, 349, 373, 397, 487, 571, 709, 787, 877, 883, 1039, 1459, 1531, 1567, 1753, 2017, 2029, 2179, 2203, 2239, 2371, 2383, 2557, 2617, 2659, 2719, 2749, 2791, 2851, 3181, 3253, 3301, 3331, 3373

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+22y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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

QuadPrimes2[1, 0, 87, 10000] (* see A106856 *)

A105389 Primes of the form x^2 + 32 y^2, also primes p with h(-p) divisible by 8.

Original entry on oeis.org

41, 113, 137, 257, 313, 337, 353, 409, 457, 521, 569, 577, 593, 761, 809, 857, 881, 953, 1129, 1153, 1201, 1217, 1249, 1321, 1553, 1601, 1657, 1777, 1889, 1993

Offset: 1

John L. Drost, May 01 2005

nonn

			41 = 9 + 32 * 1, 113 = 81 + 32 *1, 137 = 9 + 32*4

Barrucand, P. and Cohn, H. Note on primes of the form x^2 + 32 y^2, class number and residuacity, Journal fur die reine und angewandte Mathematik, v.238, pp. 67-70.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

QuadPrimes2[1, 0, 32, 10000] (* see A106856 *)
(* Second program: *)
max = 10^4; Table[yy = {y, 1, Floor[Sqrt[(max - x^2)/32]]}; Table[x^2 + 32 y^2, yy // Evaluate], {x, 1, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& (* Jean-François Alcover, Oct 04 2018 *)

A106866 Primes of the form 2x^2+xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

2, 3, 13, 41, 71, 127, 139, 179, 257, 269, 311, 331, 409, 439, 443, 461, 541, 587, 673, 739, 761, 823, 887, 967, 1051, 1061, 1087, 1129, 1153, 1223, 1237, 1277, 1373, 1381, 1409, 1427, 1439, 1567, 1619, 1733, 1741, 1783, 1933, 1973, 2017, 2063

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-23.

For even x we generate A309139. - R. J. Mathar, Aug 22 2019

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

QuadPrimes2[2, 1, 3, 10000] (* see A106856 *)

A106890 Primes of the form x^2 + xy + 11y^2, with x and y nonnegative.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 47, 53, 59, 67, 79, 83, 101, 103, 107, 109, 127, 139, 167, 181, 193, 197, 229, 239, 251, 281, 283, 293, 307, 311, 317, 337, 353, 359, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509, 541, 547, 557, 563, 569, 599, 613, 617, 619

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -43.

Rational primes that decompose in the field Q(sqrt(-43)). - N. J. A. Sloane, Dec 25 2017

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index to sequences related to decomposition of primes in quadratic fields

QuadPrimes2[1, 1, 11, 10000] (* see A106856 *)

A106915 Primes of the form 3x^2 + 2xy + 5y^2, with x and y any integer.

Original entry on oeis.org

3, 5, 13, 19, 59, 61, 83, 101, 131, 139, 157, 173, 181, 227, 229, 251, 269, 283, 293, 307, 349, 397, 419, 461, 467, 509, 523, 563, 587, 619, 643, 661, 677, 691, 733, 773, 787, 797, 811, 829, 853, 859, 941, 971, 997, 1013, 1021, 1069, 1091, 1109, 1123

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -56.
Also primes congruent to {3,5,13,19,27,45} mod 56. - Vincenzo Librandi, Jul 02 2016
The theta series for the quadratic form 3x^2 + 2xy + 5y^2 is the g.f. of A028928. - Michael Somos, Jul 02 2016
Legendre symbol (-14, a(n)) = Kronecker symbol (a(n), 14) = 1. Also, this sequence lists primes p such that Kronecker symbol (p, 2) = Legendre symbol (p, 7) = -1, i.e., primes p == 3, 5 (mod 8) and 3, 5, 6 (mod 7). - Jianing Song, Sep 04 2018

			59 is in the sequence since it is prime, and 59 = 3x^2 + 2xy + 5y^2 with x = 3 and y = 2. - _Michael B. Porter_, Jul 02 2016

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

Cf. A028928.

[p: p in PrimesUpTo(3000) | p mod 56 in {3,5,13,19,27,45}]; // Vincenzo Librandi, Jul 02 2016

Union[QuadPrimes2[3, 2, 5, 10000], QuadPrimes2[3, -2, 5, 10000]] (* see A106856 *)
Select[Prime@Range[600], MemberQ[{3, 5, 13, 19, 27, 45}, Mod[#, 56]] &] (* Vincenzo Librandi, Jul 02 2016 *)

A106919 Primes of the form 3x^2+xy+5y^2, with x and y any integer.

Original entry on oeis.org

3, 5, 7, 19, 29, 41, 53, 79, 107, 127, 137, 167, 181, 193, 199, 239, 241, 251, 257, 263, 271, 277, 281, 293, 307, 311, 331, 359, 379, 433, 449, 487, 491, 499, 523, 557, 577, 593, 599, 607, 617, 619, 643, 647, 653, 661, 709, 757, 761, 829, 853, 877, 883, 907

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-59.

Primes p such that the polynomial x^3-2x^2-1 is irreducible over Zp. The polynomial discriminant is also -59. - T. D. Noe, May 13 2005

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

Union[QuadPrimes2[3, 1, 5, 10000], QuadPrimes2[3, -1, 5, 10000]] (* see A106856 *)

cp's OEIS Frontend

A033211 Primes of form x^2 + 14*y^2.

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A033235 Primes of the form x^2 + 55*y^2.

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A033246 Primes of form x^2+71*y^2.

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A033251 Primes of form x^2+79*y^2.

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A033256 Primes of form x^2+87*y^2.

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A105389 Primes of the form x^2 + 32 y^2, also primes p with h(-p) divisible by 8.

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A106866 Primes of the form 2x^2+xy+3y^2, with x and y nonnegative.

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A106890 Primes of the form x^2 + xy + 11y^2, with x and y nonnegative.

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A106915 Primes of the form 3x^2 + 2xy + 5y^2, with x and y any integer.

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