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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 11.
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%I #12 Nov 18 2023 06:30:28

%S 1,0,0,1,2,0,2,0,1,0,1,0,0,0,0,1,0,0,2,2,0,0,0,0,3,0,0,2,0,0,0,0,0,0,

%T 4,1,2,0,0,0,0,0,2,1,2,0,0,0,3,0,0,0,2,0,2,0,0,0,0,0,0,0,2,1,0,0,0,0,

%U 0,0,0,0,0,0,0,2,2,0,2,2,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 11.

%H G. C. Greubel, <a href="/A035193/b035193.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 18 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(11, d).

%F Multiplicative with a(11^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(11, p) = -1 (p is in A296936), and a(p^e) = e+1 if Kronecker(11, p) = 1 (p is in A296935).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(3*sqrt(11)+10)/(3*sqrt(11)) = 0.60166042997... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[11, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)

%o (PARI) my(m=11); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(11, d)); \\ _Amiram Eldar_, Nov 18 2023

%Y Cf. A296935, A296936.

%K nonn,easy,mult

%O 1,5

%A _N. J. A. Sloane_