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A035160 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -30.

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%I A035160 #16 Jul 08 2025 21:30:56
%S A035160 1,1,1,1,1,1,0,1,1,1,2,1,2,0,1,1,2,1,0,1,0,2,2,1,1,2,1,0,2,1,2,1,2,2,
%T A035160 0,1,2,0,2,1,0,0,2,2,1,2,2,1,1,1,2,2,0,1,2,0,0,2,2,1,0,2,0,1,2,2,2,2,
%U A035160 2,0,0,1,0,2,1,0,0,2,2,1,1
%N A035160 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -30.
%H A035160 Amiram Eldar, <a href="/A035160/b035160.txt">Table of n, a(n) for n = 1..10000</a>
%F A035160 From _Amiram Eldar_, Nov 17 2023: (Start)
%F A035160 a(n) = Sum_{d|n} Kronecker(-30, d).
%F A035160 Multiplicative with a(p^e) = 1 if Kronecker(-30, p) = 0 (p = 2, 3 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(-30, p) = -1 (p is in A191066), and a(p^e) = e+1 if Kronecker(-30, p) = 1 (p is in A191023).
%F A035160 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(30) = 1.1471474... . (End)
%t A035160 a[n_] := DivisorSum[n, KroneckerSymbol[-30, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 17 2023 *)
%o A035160 (PARI) my(m = -30); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035160 (PARI) a(n) = sumdiv(n, d, kronecker(-30, d)); \\ _Amiram Eldar_, Nov 17 2023
%Y A035160 Cf. A191023, A191066.
%K A035160 nonn,easy,mult
%O A035160 1,11
%A A035160 _N. J. A. Sloane_