A035167 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -23.

1, 2, 2, 3, 0, 4, 0, 4, 3, 0, 0, 6, 2, 0, 0, 5, 0, 6, 0, 0, 0, 0, 1, 8, 1, 4, 4, 0, 2, 0, 2, 6, 0, 0, 0, 9, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 2, 10, 1, 2, 0, 6, 0, 8, 0, 0, 0, 4, 2, 0, 0, 4, 0, 7, 0, 0, 0, 0, 2, 0, 2, 12, 2, 0, 2, 0, 0, 8, 0, 0, 5

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• LMFDB, Dedekind zeta-function: zeta_K(s), where K is the number field with defining polynomial x^2-x+6.

Crossrefs

Cf. A191021, A191065.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -23, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 24 2021 *)

PARI

my(m = -23); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -23, d)))}; /* Michael Somos, Jan 24 2021 */

Formula

From Amiram Eldar, Nov 17 2023: (Start)

a(n) = Sum_{d|n} Kronecker(-23, d).

Multiplicative with a(23^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-23, p) = -1 (p is in A191065), and a(p^e) = e+1 if Kronecker(-23, p) = 1 (p is in A191021).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/sqrt(23) = 1.965202... . (End)