cp's OEIS Frontend

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 13. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018

From Jianing Song, Apr 21 2022: (Start)

Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.

Primes p such that p^6 == 1 (mod 13).

Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

This is an indefinite quadratic form of discriminant 13.

Also, positive integers of the form x^2+6xy-4y^2 (an indefinite quadratic form of discriminant 52).

Also, indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= 13.

From Klaus Purath, May 07 2023: (Start)

Also, positive integers of the form x^2 + (2m+1)xy + (m^2+m-3)y^2, m, x, y integers. This includes the form in the name.

Also, positive integers of the form x^2 + 2mxy + (m^2-13)y^2, m, x, y integers. This includes the form in the comment above.

This sequence contains all squares. The prime factors of the terms except for {2, 5, 7, 11, 19, ...} = A038884 are terms of the sequence. Also the products of terms belong to the sequence. Thus this set of terms is closed under multiplication.

A positive integer N belongs to the sequence if and only if N (modulo 13) is a term of A010376 and, moreover, in the case that prime factors p of N are terms of A038884, they occur only with even exponents. For these prime factors also p (modulo 13) = {2, 5, 6, 7, 8, 11} applies. (End)