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%I #33 Oct 17 2022 07:11:59
%S 1,1,1,1,2,1,2,1,1,2,2,1,0,2,2,1,0,1,0,2,2,2,0,1,3,0,1,2,2,2,2,1,2,0,
%T 4,1,0,0,0,2,0,2,0,2,2,0,0,1,3,3,0,0,2,1,4,2,0,2,2,2,0,2,2,1,0,2,0,0,
%U 0,4,0,1,2,0,3,0,4,0,2,2,1,0,2,2,0,0,2,2,0,2,0,0,2,0,0,1,2,3,2,3,2,0,2,0,4
%N a(n) = Sum_{d|n} Kronecker(-6, d).
%H G. C. Greubel, <a href="/A192013/b192013.txt">Table of n, a(n) for n = 1..1000</a>
%F Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...].
%F a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
%F G.f.: Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)).
%F Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-6, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-6, p) * p^-s)).
%F a(n) = A000377(n) = A000377(2*n) = A190611(2*n). a(n) = a(2*n) = a(3*n). - _Michael Somos_, Jul 22 2015
%F 0 <= a(n) <= d(n) and these bounds are sharp. - _Charles R Greathouse IV_, Dec 14 2016
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - _Amiram Eldar_, Oct 17 2022
%e G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ...
%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -6, d], { d, Divisors[n]}]];
%o (PARI) {a(n) = sumdiv( n, d, kronecker( -6, d))};
%o (PARI) a(n)=sumdivmult(n, d, kronecker(-6, d)) \\ _Charles R Greathouse IV_, Dec 14 2016
%Y Cf. A000377, A190611.
%Y Cf. A109017(n) = Kronecker(-6, n). - _Michael Somos_, Jul 22 2015
%K nonn,mult
%O 1,5
%A _Michael Somos_, Jun 22 2011
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