A123064 -id:A123064 - cp's OEIS frontend

A106872 Primes of the form 2x^2+xy+4y^2.

2, 5, 7, 19, 41, 59, 71, 97, 101, 103, 107, 109, 113, 157, 163, 191, 193, 211, 233, 257, 281, 307, 311, 317, 359, 373, 397, 419, 421, 439, 443, 467, 479, 503, 541, 547, 563, 593, 599, 659, 661, 683, 691, 701, 727, 733, 751, 769, 877, 887, 907, 977, 997, 1033

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-31.

Primes p such that the polynomial x^3-x^2-1 is irreducible over Zp. The polynomial discriminant is also -31. - 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)

Primes in A123064.

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

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).

A123065 Numbers primitively represented by the quadratic form 2 x^2 + xy + 4 y^2.

Original entry on oeis.org

2, 4, 5, 7, 10, 14, 16, 19, 20, 25, 28, 32

Offset: 1

N. J. A. Sloane, Sep 27 2006

nonn

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

Cf. A123063, A123064, A123067, A123068.

A123067 Theta series of the "Little Methuselah" quadratic form x^2 + 2y^2 + yz + 4*z^2.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 8, 2, 10, 10, 2, 12, 4, 4, 10, 4, 10, 12, 10, 10, 20, 12, 4, 12, 12, 12, 8, 8, 8, 24, 8, 0, 24, 12, 8, 10, 24, 12, 6, 12, 14, 38, 8, 8, 32, 18, 8, 4, 8, 20, 20, 16, 12, 16, 24, 4, 30, 28, 4, 14, 20, 12, 2, 18, 18, 44, 20, 8, 28, 12, 10, 26, 30, 12, 28, 16, 20, 20, 8, 16, 34

Offset: 0

N. J. A. Sloane, Sep 27 2006, corrected Oct 26 2006

nonn

The Little Methuselah form represents every integer from 1 to 30, but fails to represent 31. Every integer-valued positive definite ternary form not equivalent to it fails to represent some integer between 1 and 30. [Conway]

			1 + 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 2*x^7 + 10*x^8 + 10*x^9 + 2*x^10 + ...

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, pp. 81-82.

Cf. A123063, A123064, A123065, A123068.

L:=LatticeWithGram(3, [2,0,0,0,4,1,0,1,8] ); T := ThetaSeries(L,500); T;

{a(n)= if(n<1, n==0, qfrep([2, 0, 0; 0, 4, 1; 0, 1, 8],n, 1)[n]*2)} /* Michael Somos, Oct 23 2006 */

A123068 Numbers represented by the "Little Methuselah" quadratic form x^2 + 2y^2 + yz + 4*z^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

N. J. A. Sloane, Sep 27 2006

nonn

Theorem (Conway, p. 81) This ternary form represents every number from 0 to 32 except 31. Any other integer-valued ternary form not equivalent to this one fails to represent some number between 1 and 30.

			1+2*x+2*x^2+4*x^3+4*x^4+6*x^5+8*x^6+2*x^7+10*x^8+10*x^9+2*x^10+...

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

Cf. A123063, A123064, A123065, A123067.

L:=LatticeWithGram(3, [2,0,0,0,4,1,0,1,8] ); T := ThetaSeries(L,500); T;

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 *)

cp's OEIS Frontend

A106872 Primes of the form 2x^2+xy+4y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Magma

Magma

Mathematica

PARI

Formula

A123065 Numbers primitively represented by the quadratic form 2 x^2 + xy + 4 y^2.

Views

Author

Keywords

References

Crossrefs

A123067 Theta series of the "Little Methuselah" quadratic form x^2 + 2*y^2 + y*z + 4*z^2.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Magma

PARI

A123068 Numbers represented by the "Little Methuselah" quadratic form x^2 + 2*y^2 + y*z + 4*z^2.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Magma

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A123067 Theta series of the "Little Methuselah" quadratic form x^2 + 2y^2 + yz + 4*z^2.

A123068 Numbers represented by the "Little Methuselah" quadratic form x^2 + 2y^2 + yz + 4*z^2.