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%I #17 Nov 20 2023 11:53:04
%S 1,0,1,1,0,0,0,0,1,0,2,1,2,0,0,1,0,0,0,0,0,0,2,0,1,0,1,0,0,0,0,0,2,0,
%T 0,1,2,0,2,0,0,0,0,2,0,0,2,1,1,0,0,2,0,0,0,0,0,0,2,0,2,0,0,1,0,0,0,0,
%U 2,0,2,0,2,0,1,0,0,0,0,0,1
%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 3.
%H G. C. Greubel, <a href="/A035186/b035186.txt">Table of n, a(n) for n = 1..10000</a>
%F From _Amiram Eldar_, Oct 17 2022: (Start)
%F a(n) = Sum_{d|n} Kronecker(3, d).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2+sqrt(3)) / (3*sqrt(3)) = 0.506897... . (End)
%F Multiplicative with a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(3, p) = -1 (p is in A038875), and a(p^e) = e+1 if Kronecker(3, p) = 1 (p is in A097933). - _Amiram Eldar_, Nov 20 2023
%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[3, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)
%o (PARI) my(m=3); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o (PARI) a(n) = sumdiv(n, d, kronecker(3, d)); \\ _Amiram Eldar_, Nov 20 2023
%Y Cf. A038875, A097933.
%K nonn,easy,mult
%O 1,11
%A _N. J. A. Sloane_
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