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A099379 The real part of n', the arithmetic derivative for Gaussian integers. 7
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; text; internal format)
OFFSET
0,3
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 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.
LINKS
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Eric Weisstein's World of Mathematics, Gaussian Integer
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).
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.
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}]]
CROSSREFS
Cf. A003415 (arithmetic derivative of n), A099380 (imaginary part of the Gaussian-integer derivative of n).
Cf. A218854, A218855 (triangles of derivatives of Gaussian integers).
Cf. A218856, A218857 (triangles of derivatives of Gaussian integers).
Sequence in context: A258247 A085470 A200584 * A234014 A367994 A208931
KEYWORD
nice,nonn
AUTHOR
T. D. Noe, Oct 14 2004
STATUS
approved

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Last modified June 24 15:31 EDT 2024. Contains 373679 sequences. (Running on oeis4.)