A218857 Imaginary part of the arithmetic derivative for the triangle of Gaussian integers z = r + i*I.

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, 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, -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

Offset

0,6

Comments

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

Links

T. D. Noe, Rows n = 0..50 of triangle, flattened

Example

Triangle:
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, 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, -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

Mathematica

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}]

Crossrefs

Cf. A099379, A099380.

Sequence in context: A263452 A133457 A324120 * A260803 A260804 A353698

Adjacent sequences: A218854 A218855 A218856 * A218858 A218859 A218860

Keyword

sign,tabf

Author

T. D. Noe, Nov 12 2012

Status

approved