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%I #21 Dec 25 2016 00:38:35
%S 0,0,2,1,8,3,8,1,24,6,16,1,28,5,16,14,64,5,30,1,52,10,24,1,80,30,36,
%T 27,60,7,58,1,160,14,44,26,96,7,40,28,144,9,62,1,92,57,48,1,208,14,
%U 110,32,124,9,108,38,176,22,72,1,176,11,64,51,384,64,94,1,156,26,122,1,264,11
%N The real part of n', the arithmetic derivative for Gaussian integers.
%C Ufnarovski and Ahlander briefly mention this idea, but they do not pursue it because the derivative of Gaussian integers is not an extension of the arithmetic derivative of integers. Recall that every nonzero Gaussian integer has a unique factorization into the product of a unit (1, -1, i, -i) and powers of positive Gaussian primes (i.e., Gaussian primes a+bi with a>0 and b>=0). The derivative of all positive Gaussian primes is 1. The derivative of 0 or a unit is 0. The derivative of a product follows the Leibniz rule (uv)' = uv' + vu'. Note that (-u)' = -(u') and (iu)' = i(u'). This definition of a derivative can be extended to fractions u/v, where u and v are Gaussian integers. Indeed, the Mathematica code shown here works with such fractions.
%H T. D. Noe, <a href="/A099379/b099379.txt">Table of n, a(n) for n = 0..2048</a>
%H Victor Ufnarovski and Bo Ahlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaussianInteger.html">Gaussian Integer</a>
%F If n = u Product p_i^e_i, where the p_i are positive Gaussian primes and u is a unit, then a(n) = n * Sum (e_i/p_i).
%e For n=5, the factorization into positive Gaussian integers is -i (1+2i) (2+i). Using the formula, the derivative is 5 (1/(1+2i) + 1/(2+i)) = 3-3i. Hence a(5) = 3.
%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]])]; Re[Table[di[n], {n, 0, 100}]]
%Y Cf. A003415 (arithmetic derivative of n), A099380 (imaginary part of the Gaussian-integer derivative of n).
%Y Cf. A218854, A218855 (triangles of derivatives of Gaussian integers).
%Y Cf. A218856, A218857 (triangles of derivatives of Gaussian integers).
%K nice,nonn
%O 0,3
%A _T. D. Noe_, Oct 14 2004
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