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A099379
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The real part of n', the arithmetic derivative for Gaussian integers.
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2
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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, 60, 7, 58, 1, 160, 14, 44, 26, 96, 7, 40, 28, 144, 9, 62, 1, 92, 57, 48, 1, 208, 14, 110, 32, 124, 9, 108, 38, 176, 22, 72, 1, 176, 11, 64, 51, 384, 64, 94, 1, 156, 26, 122, 1, 264, 11
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..2048
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Eric Weisstein's World of Mathematics, Gaussian Integer
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FORMULA
| 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).
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EXAMPLE
| 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.
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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]])]; Re[Table[di[n], {n, 0, 100}]]
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CROSSREFS
| Cf. A003415 (arithmetic derivative of n), A099380 (imaginary part of the Gaussian-integer derivative of n).
Sequence in context: A075647 A085470 A200584 * A133214 A191935 A142075
Adjacent sequences: A099376 A099377 A099378 * A099380 A099381 A099382
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KEYWORD
| nice,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Oct 14 2004
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