A106872 -id:A106872 - cp's OEIS frontend

A123064 Numbers represented by the quadratic form 2 x^2 + xy + 4 y^2.

0, 2, 4, 5, 7, 8, 10, 14, 16, 18, 19, 20, 25, 28, 32, 35, 36, 38, 40, 41, 45, 49, 50, 56, 59, 62, 63, 64, 70, 71, 72, 76, 80, 82, 90, 94, 95, 97, 98, 100, 101, 103, 107, 109, 112, 113, 118, 124, 125, 126, 128, 133, 134, 140, 142, 144, 152, 155, 157, 160, 162, 163, 164, 171, 175

Offset: 1

N. J. A. Sloane, Sep 27 2006

nonn

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 82.

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

Cf. A123063, A123065, A123067, A123068.

Primes: A106872.

L:=LatticeWithGram(2, [4,1,1,8] ); T := ThetaSeries(L,500); T;

A123063 Theta series of lattice with Gram matrix [4,1;1,8].

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 2, 4, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 4, 0, 0, 2, 2, 0, 2, 0, 6, 2, 0, 0, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 6, 0, 0, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 4, 0, 6, 2, 0, 2, 0

Offset: 0

N. J. A. Sloane, Sep 27 2006

nonn

a(n) = number of solutions to n = 2*x^2 + x*y + 4*y^2 in integers, hence a(n) nonzero if and only if n is in A123064 and p is prime and a(p) = 2 if and only if p is in A106872. - Michael Somos, Jul 16 2011

			G.f. = 1 + 2*x^2 + 2*x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^14 + 4*x^16 + 2*x^18 + ...
G.f. =  1 + 2*q^4 + 2*q^8 + 2*q^10 + 2*q^14 + 2*q^16 + 2*q^20 + 2*q^28 + 4*q^32 + ...

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 82.

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

Cf. A106872, A123064, A123065.

L:=LatticeWithGram(2, [4,1,1,8] ); T := ThetaSeries(L,500); T;

A := Basis( ModularForms( Gamma1(31), 1), 103); A[1] + 2*A[3] + 2*A[5] + 2*A[6] + 2*A[8] + 2*A[9] + 2*A[11] + 2*A[15]; /* Michael Somos, Jun 14 2014 */

a := func ; /* Michael Somos, Jun 14 2014 */

terms = 105; max = terms+3; s = Sum[x^(2*n^2 + n*m + 4*m^2), {n, -max, max}, {m, -max, max}] + O[x]^max; CoefficientList[s, x][[1 ;; terms]] (* Jean-François Alcover, Jul 05 2017 *)

{a(n) = if( n<1, n==0, qfrep( [4, 1; 1, 8], n, 1)[n] * 2)}; /* Michael Somos, Sep 28 2006 */

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6*u1^3 + 2*u3*u2^3 - 3*u3^3*u2 - 6*u6^3*u1 + 6*u6*u2^2*u1 - 6*u3*u2^2*u1 + 3*u3*u2*u1^2 - 6*u6*u2*u1^2 - 9*u6*u3^2*u1 - 18*u6^2*u3*u2 + 18*u6*u3^2*u2 + 18*u6^2*u3*u1. - Michael Somos, Sep 28 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (31 t)) = 31^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 16 2011
G.f.: Sum_{n,m in Z} x^(2*n^2 + n*m + 4*m^2).

A106871 Primes of the form 2x^2+xy+4y^2, with x and y nonnegative.

Original entry on oeis.org

2, 7, 41, 59, 101, 107, 109, 233, 257, 281, 307, 311, 397, 419, 421, 503, 599, 659, 691, 733, 751, 907, 1051, 1061, 1087, 1103, 1163, 1217, 1249, 1279, 1291, 1307, 1321, 1361, 1373, 1433, 1471, 1489, 1597, 1621, 1693, 1733, 1777, 1787, 1831

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-31.

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. A123064, A106872.

QuadPrimes2[2, 1, 4, 10000] (* see A106856 *)

A106873 Primes of the form 2x^2-xy+4y^2, with x and y nonnegative.

Original entry on oeis.org

2, 5, 19, 71, 97, 103, 113, 157, 163, 191, 193, 211, 317, 359, 373, 439, 443, 467, 479, 541, 547, 563, 593, 661, 683, 701, 727, 769, 877, 887, 977, 997, 1033, 1039, 1093, 1123, 1151, 1223, 1229, 1289, 1399, 1409, 1423, 1427, 1459, 1601, 1609, 1613

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-31.

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. A123064, A106872.

QuadPrimes2[2, -1, 4, 10000] (* see A106856 *)

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

Original entry on oeis.org

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset: 1

T. D. Noe, May 19 2005

Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

			For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

Eric Weisstein's World of Mathematics, Class Number

Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).

cp's OEIS Frontend

A123064 Numbers represented by the quadratic form 2 x^2 + xy + 4 y^2.

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A123063 Theta series of lattice with Gram matrix [4,1;1,8].

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A106871 Primes of the form 2x^2+xy+4y^2, with x and y nonnegative.

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A106873 Primes of the form 2x^2-xy+4y^2, with x and y nonnegative.

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Mathematica

A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

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