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A103226
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Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.
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2
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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, -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, 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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
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LINKS
| Eric Weisstein's World of Mathematics, Moebius Function
T. D. Noe, Plot of the Moebius function for Gaussian Integers
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EXAMPLE
| The table is symmetric and begins
0 1 0 -1 0 1 0 -1 0 0 0
1 -1 -1 1 -1 1 -1 0 1 1 -1
0 -1 0 -1 0 -1 0 -1 0 1 0
-1 1 -1 1 0 1 1 1 -1 -1 -1
0 -1 0 0 0 -1 0 1 0 -1 0
1 1 -1 1 -1 -1 -1 1 -1 1 0
0 -1 0 1 0 -1 0 1 0 1 0
-1 0 -1 1 1 1 1 1 -1 -1 -1
0 1 0 -1 0 -1 0 -1 0 1 0
0 1 1 -1 -1 1 1 -1 1 0 -1
0 -1 0 -1 0 0 0 -1 0 -1 0
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MATHEMATICA
| 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}]]
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CROSSREFS
| Cf. A103227 (least k such that (2n-1)+ki is squarefull).
Sequence in context: A048820 A144101 A127000 * A080886 A083924 A171587
Adjacent sequences: A103223 A103224 A103225 * A103227 A103228 A103229
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KEYWORD
| nice,sign,tabl
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jan 26 2005
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