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%I #10 Feb 16 2025 08:32:56
%S 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 -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 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 Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.
%C 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 T. D. Noe, <a href="http://www.sspectra.com/math/ComplexMoebius.gif">Plot of the Moebius function for Gaussian Integers</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%e The table is symmetric and begins
%e 0 1 0 -1 0 1 0 -1 0 0 0
%e 1 -1 -1 1 -1 1 -1 0 1 1 -1
%e 0 -1 0 -1 0 -1 0 -1 0 1 0
%e -1 1 -1 1 0 1 1 1 -1 -1 -1
%e 0 -1 0 0 0 -1 0 1 0 -1 0
%e 1 1 -1 1 -1 -1 -1 1 -1 1 0
%e 0 -1 0 1 0 -1 0 1 0 1 0
%e -1 0 -1 1 1 1 1 1 -1 -1 -1
%e 0 1 0 -1 0 -1 0 -1 0 1 0
%e 0 1 1 -1 -1 1 1 -1 1 0 -1
%e 0 -1 0 -1 0 0 0 -1 0 -1 0
%t 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 Cf. A103227 (least k such that (2n-1)+ki is squarefull).
%K nice,sign,tabl,changed
%O 1,1
%A _T. D. Noe_, Jan 26 2005