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Imaginary part of the arithmetic derivative for the triangle of Gaussian integers z = r + i*I.
3

%I #6 Nov 12 2012 02:37:44

%S 0,0,0,0,0,-2,0,2,1,2,0,-2,-1,0,0,0,1,1,1,0,0,0,-1,-1,-1,-8,-2,0,2,8,

%T 3,6,3,8,2,0,-2,-8,-3,-6,-3,-3,0,0,0,0,3,1,1,1,1,3,0,0,0,0,-3,-1,-1,

%U -1,-1,-6,-3,-2,1,2,3,8,4,8,4,8,4,6,3,2,-1,-2,-3,-8,-4,-8,-4,-8,-4

%N Imaginary part of the arithmetic derivative for the triangle of Gaussian integers z = r + i*I.

%C The real part is A218856, which has more information, including a plot. Consult A099379 for the arithmetic derivative of Gaussian integers.

%H T. D. Noe, <a href="/A218857/b218857.txt">Rows n = 0..50 of triangle, flattened</a>

%e Triangle:

%e 0,

%e 0, 0, 0, 0,

%e -2, 0, 2, 1, 2, 0, -2, -1,

%e 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1, -1,

%e -8, -2, 0, 2, 8, 3, 6, 3, 8, 2, 0, -2, -8, -3, -6, -3,

%e -3, 0, 0, 0, 0, 3, 1, 1, 1, 1, 3, 0, 0, 0, 0, -3, -1, -1, -1, -1,

%e -6, -3, -2, 1, 2, 3, 8, 4, 8, 4, 8, 4, 6, 3, 2, -1, -2, -3, -8, -4, -8, -4, -8, -4

%t di[0]=0; di[1]=0; di[ -1]=0; di[I]=0; di[ -I]=0; di[n_] := Module[{f, unt}, f=FactorInteger[n, GaussianIntegers->True]; unt=(Abs[f[[1, 1]]]==1); If[unt, f=Delete[f, 1]]; f=Transpose[f]; Plus@@(n*f[[2]]/f[[1]])]; Table[t = Join[Table[di[n - i + I*i], {i, 0, n}], Table[di[i - n + I*i], {i, n - 1, 0, -1}], Table[di[i - n - I*i], {i, 1, n}], Table[di[n - i - I*i], {i, n - 1, 1, -1}]]; Im[t], {n, 0, 6}]

%Y Cf. A099379, A099380.

%K sign,tabf

%O 0,6

%A _T. D. Noe_, Nov 12 2012