%I #15 Oct 27 2023 19:27:09

%S 0,0,0,0,4,0,0,0,32,1,0,0,80,0,5,16,176,0,7,0,48,1,0,0,112,1,12,27,

%T 176,0,0,0,368,6,0,32,96,0,7,80,156,0,0,0,240,32,7,0,608,6,16,44,96,0,

%U 216,80,272,1,0,0,272,0,9,24,2368,10,0,0,220,8,0,0,284,0

%N n''', the third arithmetic derivative of n.

%C For prime p, a(p^p) = p^p.

%C a(A157037(n)) = 0. - _Reinhard Zumkeller_, Feb 22 2009

%D See A003415.

%H Alois P. Heinz, <a href="/A099306/b099306.txt">Table of n, a(n) for n = 0..10000</a> (first 2000 terms from T. D. Noe)

%F a(n) = A003415(A003415(A003415(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[dn[dn[n]]], {n, 100}]

%Y Cf. A003415 (arithmetic derivative of n), A068346 (second arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero).

%Y Column k=3 of A258651.

%K nonn

%O 0,5

%A _T. D. Noe_, Oct 12 2004