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A229341 a(n) = tau(n'), the number of divisors of the arithmetic derivative of n.

%I A229341 #49 Mar 26 2018 20:59:27
%S A229341 1,1,3,1,2,1,6,4,2,1,5,1,3,4,6,1,4,1,8,4,2,1,6,4,4,4,6,1,2,1,10,4,2,6,
%T A229341 12,1,4,5,6,1,2,1,10,4,3,1,10,4,6,6,8,1,5,5,6,4,2,1,6,1,4,4,14,6,2,1,
%U A229341 12,4,2,1,12,1,4,4,10,6,2,1,10,12,2,1,6,4,6
%N A229341 a(n) = tau(n'), the number of divisors of the arithmetic derivative of n.
%H A229341 Antti Karttunen, <a href="/A229341/b229341.txt">Table of n, a(n) for n = 2..65537</a>
%F A229341 a(n) = A000005(A003415(n)).
%e A229341 For n=4, tau(n')=tau(4)=3.
%e A229341 For n=5, tau(n')=tau(1)=1.
%t A229341 dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose@ FactorInteger@ n}, If[PrimeQ@n, 1, Total[n*f[[2]]/f[[1]]]]]; (* see A003415 *); f[n_] := DivisorSigma[0, dn@ n]; Array[f, 85, 2] (* _Robert G. Wilson v_, Mar 12 2018 *)
%o A229341 (PARI) rd(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1];));}
%o A229341 a(n) = numdiv(rd(n)); \\ _Michel Marcus_, Sep 24 2013
%o A229341 (GAP) List(List(List([2..10^2],Factors),i->Product(i)*Sum(i,j->1/j)),Tau); # _Muniru A Asiru_, Mar 05 2018
%Y A229341 Cf. A000005, A003415.
%K A229341 nonn
%O A229341 2,3
%A A229341 _Luca Brigada Villa_, Sep 24 2013