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A103226 Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.

%I A103226 #10 Feb 16 2025 08:32:56
%S A103226 0,1,1,0,-1,0,-1,-1,-1,-1,0,1,0,1,0,1,-1,-1,-1,-1,1,0,1,0,1,0,1,0,-1,
%T A103226 -1,-1,0,0,-1,-1,-1,0,0,0,1,0,1,0,0,0,0,1,-1,1,-1,-1,1,-1,1,0,0,1,0,1,
%U A103226 0,-1,0,1,0,1,0,-1,-1,1,-1,1,-1,-1,1,-1,1,-1,-1,0,1,0,-1,0,1,0,1,0,-1,0,1,0,1,1,0,-1,-1,-1,1,1,-1,-1,-1,0,1,1,0,-1,0
%N A103226 Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.
%C A103226 The usual definition of the Moebius function is used, except that Gaussian primes are used instead of rational primes. Consider the diagonal (a-b)+bi of Gaussian integers for 0<=b<=a. It appears that the diagonals for a=1, 3, 5 and 11 are the only ones containing just -1 and 1; these Gaussian integers are all squarefree. Interestingly, as shown in A103227, for all n there is some 0<=k<=12 such that n+ki is a squarefull Gaussian integer.
%H A103226 T. D. Noe, <a href="http://www.sspectra.com/math/ComplexMoebius.gif">Plot of the Moebius function for Gaussian Integers</a>
%H A103226 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%e A103226 The table is symmetric and begins
%e A103226    0  1  0 -1  0  1  0 -1  0  0  0
%e A103226    1 -1 -1  1 -1  1 -1  0  1  1 -1
%e A103226    0 -1  0 -1  0 -1  0 -1  0  1  0
%e A103226   -1  1 -1  1  0  1  1  1 -1 -1 -1
%e A103226    0 -1  0  0  0 -1  0  1  0 -1  0
%e A103226    1  1 -1  1 -1 -1 -1  1 -1  1  0
%e A103226    0 -1  0  1  0 -1  0  1  0  1  0
%e A103226   -1  0 -1  1  1  1  1  1 -1 -1 -1
%e A103226    0  1  0 -1  0 -1  0 -1  0  1  0
%e A103226    0  1  1 -1 -1  1  1 -1  1  0 -1
%e A103226    0 -1  0 -1  0  0  0 -1  0 -1  0
%t A103226 moebius[z_] := Module[{f, mu}, If[z==0, mu=0, If[Abs[z]==1, mu=1, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, f=Drop[f, 1]]; mu=1; Do[If[f[[i, 2]]==1, mu=-mu, mu=0], {i, Length[f]}]]]; mu]; Flatten[Table[z=(n-k)+k*I; moebius[z], {n, 0, 15}, {k, 0, n}]]
%Y A103226 Cf. A103227 (least k such that (2n-1)+ki is squarefull).
%K A103226 nice,sign,tabl
%O A103226 1,1
%A A103226 _T. D. Noe_, Jan 26 2005