A106915 -id:A106915 - cp's OEIS frontend

A028928 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].

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

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The number of integer solutions (x, y) to n = 3*x^2 + 2*x*y + 5*y^2, discriminant -56. - Ray Chandler, Jul 12 2014

			G.f. = 1 + 2*q^3 + 2*q^5 + 2*q^6 + 2*q^10 + 2*q^12 + 2*q^13 + 2*q^19 + 2*q^20 + 2*q^21 + 2*q^24 + 2*q^26 + 4*q^27 + 2*q^35 + 2*q^38 + 2*q^40 + 2*q^42 + 6*q^45 + ...

John Cannon, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A106915.

a[ n_] := If[ n < 1, Boole[ n == 0], If[ -1 != KroneckerSymbol[ -7, n / 7^IntegerExponent[ n, 7]], 0, Sum[ KroneckerSymbol[ -14, d], { d, Divisors @ n}]]]; (* Michael Somos, Jul 13 2011 *)

{a(n) = if( n<1, n==0, qfrep([3, 1; 1, 5], n)[n] * 2)}; /* Michael Somos, Jun 06 2011 */

{a(n) = if( n<1, n==0, (-1 == kronecker( -7, n / 7^valuation( n, 7))) * sumdiv( n, d, kronecker( -14, d)))}; /* Michael Somos, Jun 22 2011 */

Expansion of phi(q^3) * phi(q^42) + 2*q^5 * chi(q) * psi(-q^3) * chi(q^14) * psi(-q^42) = phi(q^6) * phi(q^21) + 2*q^3 * chi(q^2) * psi(-q^6) * chi(q^7) * psi(-q^21) = phi(q^2) * phi(q^7) - 2*q^2 * phi(-q^4) * psi(q^7) * chi(-q) / chi(-q^28) in powers of q where phi(), psi(), chi() are Ramanujan theta functions - Michael Somos and Alex Berkovich, Jun 06 2011
Expansion of - phi(q) * phi(q^14) + 2 * chi(q) * f(-q^7) * f(-q^8) * chi(q^14) in powers of q where phi(), chi(), f() are Ramanujan theta functions - Michael Somos, Jun 22 2011
G.f.: Sum_{n, m in Z} x^(3*n*n + 2*n*m + 5*m*m).

A274675 Primes p such that p = x^2 + 14y^2 or p = 2x^2 + 7*y^2, where p != 2, 7 and x, y are integers.

Original entry on oeis.org

23, 71, 79, 113, 127, 137, 151, 191, 193, 233, 239, 263, 281, 337, 359, 401, 431, 449, 457, 463, 487, 569, 599, 617, 631, 641, 673, 743, 751, 809, 823, 863, 911, 919, 953, 967, 977, 991, 1009, 1031, 1033, 1087, 1103, 1129, 1201, 1289, 1297, 1303, 1327

Offset: 1

Vincenzo Librandi, Jul 02 2016

Also primes congruent to {1, 9, 15, 23, 25, 39} mod 56.
From Wolfdieter Lang, Jun 04 2021: (Start)
Discriminant -8*7. The product of two entries is congruent to {1, 7} (mod 8). (Buell, p. 51, 3).
The given two reduced positive definite binary quadratic forms represent the odd primes, not 7, with the generic characters Legendre(p|7) = +1 and Legendre(2|p) = +1. The other two reduced forms are [3, 2, 5] and [3,-2, 5] with values -1 and -1 for these two generic characters, and give the odd primes, not 7, listed in A106915. This is related to the two genera of discriminant -56 with class number h(-56) = 4. See Buell, p. 52, 2), and Cox, p. 30.
There is a misprint 29 (instead of 39) in Cox (1989, ISBN 0-471-50654-0), p. 33, in eqs. (2.21) and (2.23). (End)
In the first US edition, there just one error, in Eq. (2.21), and it is on page 33. In the second edition this error has been corrected. - N. J. A. Sloane, Jun 04 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 51-53.
David A. Cox, Primes of the Form x^2 + n y^2, John Wiley & Sons, 1st edition, 1989; 2nd edition, 2003.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas R. Hagedorn, Primes of the form x^2+ny^2 and the geometry of (convenient) numbers (page 3).

Cf. A033202, A106915.

[p: p in PrimesUpTo(3000) | p mod 56 in {1, 9, 15, 23, 25, 39} ];

Select[Prime@Range[300], MemberQ[{1, 9, 15, 23, 25, 39}, Mod[#, 56]] &]

is(n) = ispseudoprime(n) && #setintersect([n % 56], [1, 9, 15, 23, 25, 39])==1 \\ Felix Fröhlich, Jul 02 2016

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A028928 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].

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A274675 Primes p such that p = x^2 + 14y^2 or p = 2x^2 + 7*y^2, where p != 2, 7 and x, y are integers.

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cp's OEIS Frontend

A028928 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A274675 Primes p such that p = x^2 + 14*y^2 or p = 2*x^2 + 7*y^2, where p != 2, 7 and x, y are integers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A274675 Primes p such that p = x^2 + 14y^2 or p = 2x^2 + 7*y^2, where p != 2, 7 and x, y are integers.