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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 = -10.
2

%I #13 Nov 17 2023 07:34:43

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

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

%U 0,2,0,1,0,2,0,2,4,0,0,1,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -10.

%H G. C. Greubel, <a href="/A035180/b035180.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 17 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(-10, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(-10, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(-10, p) = -1 (p is in A296925), and a(p^e) = e+1 if Kronecker(-10, p) = 1.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(10) = 0.993458... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-10, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)

%o (PARI) my(m=-10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-10, d)); \\ _Amiram Eldar_, Nov 17 2023

%Y Cf. A296925.

%K nonn,easy,mult

%O 1,7

%A _N. J. A. Sloane_