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%I #19 Nov 17 2023 17:51:23
%S 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,
%T 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,
%U 2,0,2,12,2,0,2,0,0,8,0,0,5
%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -23.
%H Amiram Eldar, <a href="/A035167/b035167.txt">Table of n, a(n) for n = 1..10000</a>
%H LMFDB, <a href="https://www.lmfdb.org/L/NumberField/2.0.23.1/">Dedekind zeta-function: zeta_K(s), where K is the number field with defining polynomial x^2-x+6</a>.
%F From _Amiram Eldar_, Nov 17 2023: (Start)
%F a(n) = Sum_{d|n} Kronecker(-23, d).
%F 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).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/sqrt(23) = 1.965202... . (End)
%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -23, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Jan 24 2021 *)
%o (PARI) my(m = -23); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -23, d)))}; /* _Michael Somos_, Jan 24 2021 */
%Y Cf. A191021, A191065.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_.
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