%I A099379
%S A099379 0,0,2,1,8,3,8,1,24,6,16,1,28,5,16,14,64,5,30,1,52,10,24,1,80,30,36,27,
%T A099379 60,7,58,1,160,14,44,26,96,7,40,28,144,9,62,1,92,57,48,1,208,14,110,32,
%U A099379 124,9,108,38,176,22,72,1,176,11,64,51,384,64,94,1,156,26,122,1,264,11
%N A099379 The real part of n', the arithmetic derivative for Gaussian integers.
%C A099379 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 Leibnitz 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 A099379 T. D. Noe, <a href="b099379.txt">Table of n, a(n) for n=0..2048</a>
%H A099379 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">How to Differentiate a Number</a>, J. Integer 
               Seqs., Vol. 6, 2003.
%H A099379 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GaussianInteger.html">Gaussian Integer</a>
%F A099379 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 A099379 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 A099379 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 A099379 Cf. A003415 (arithmetic derivative of n), A099380 (imaginary part of 
               the Gaussian-integer derivative of n).
%Y A099379 Sequence in context: A082834 A075647 A085470 this_sequence A133214 A142075 
               A156365
%Y A099379 Adjacent sequences: A099376 A099377 A099378 this_sequence A099380 A099381 
               A099382
%K A099379 nice,nonn
%O A099379 0,3
%A A099379 T. D. Noe (noe(AT)sspectra.com), Oct 14 2004