|
| |
|
|
A103226
|
|
Moebius function mu(n+ki) defined for the Gaussian integers. The table begins with n=k=0 and is read by antidiagonals.
|
|
3
|
|
|
|
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;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
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.
|
|
|
LINKS
|
|
|
|
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
|
|
|
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}]]
|
|
|
CROSSREFS
|
Cf. A103227 (least k such that (2n-1)+ki is squarefull).
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
