A099310 - OEIS

%I #9 Sep 30 2017 23:47:54

%S 0,0,1,1,4,1,5,4,5,4,7,4,16,5,12,12,32,5,21,12,16,7,13,12,24,16,21,16,

%T 32,12,31,32,24,32,44,16,60,21,44,32,68,16,41,24,44,13,25,32,41,24,80,

%U 44,56,21,68,44,60,32,31,32,92,31,60,80,112,24,61,80,48,44,59,44,156,60

%N Arithmetic derivative of Euler's totient function phi(n).

%D See A003415

%H Michel Marcus, <a href="/A099310/b099310.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = A003415(A000010(n))

%t dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[EulerPhi[n]], {n, 100}]

%o (GAP)

%o A099310:= Concatenation([0,0],List(List(List([3..10^3], n->Phi(n)),Factors),i->Product(i)*Sum(i,j->1/j))); # _Muniru A Asiru_, Sep 27 2017

%o (PARI) ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);

%o a(n) = ad(eulerphi(n)); \\ _Michel Marcus_, Sep 29 2017

%Y Cf. A003415 (arithmetic derivative of n).

%K nonn

%O 1,5

%A _T. D. Noe_, Oct 12 2004