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%I #16 Nov 20 2023 11:53:11
%S 1,2,2,3,0,4,1,4,3,0,0,6,0,2,0,5,0,6,2,0,2,0,0,8,1,0,4,3,2,0,2,6,0,0,
%T 0,9,2,4,0,0,0,4,0,0,0,0,2,10,1,2,0,0,2,8,0,4,4,4,2,0,0,4,3,7,0,0,0,0,
%U 0,0,0,12,0,4,2,6,0,0,0,0,5
%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 7.
%H G. C. Greubel, <a href="/A035189/b035189.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(7, d).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(8+3*sqrt(7)) / sqrt(7) = 2.0929097... . (End)
%F Multiplicative with a(7^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(7, p) = -1 (p is in A003632), and a(p^e) = e+1 if Kronecker(7, p) = 1 (p is in A038878 \ {7}). - _Amiram Eldar_, Nov 20 2023
%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[7, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)
%o (PARI) my(m=7); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o (PARI) a(n) = sumdiv(n, d, kronecker(7, d)); \\ _Amiram Eldar_, Nov 20 2023
%Y Cf. A003632, A038878.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_
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