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A231864 Partial sums of the second power of arithmetic derivative function A003415.

%I A231864 #15 Nov 21 2013 18:19:27
%S A231864 0,1,2,18,19,44,45,189,225,274,275,531,532,613,677,1701,1702,2143,
%T A231864 2144,2720,2820,2989,2990,4926,5026,5251,5980,7004,7005,7966,7967,
%U A231864 14367,14563,14924,15068,18668,18669,19110,19366,23990,23991,25672,25673,27977,29498
%N A231864 Partial sums of the second power of arithmetic derivative function A003415.
%C A231864 a(n)-> ~ 0.4*n^3 as n-> oo (note: 1^2+2^2+3^3+4^4+5^4 ...-> ~ 1/3*n^3)
%C A231864 Note: the partial sums of a power of the arithmetic derivatives of the natural numbers tend to infinity as the partial sums of the natural numbers of the same power.  In more general sense: sum(D^d(i)^m, i = 1..n) -> k*n^(m+1) as n-> oo where D^d(i) is the derivative of order d th of the natural number i (d may be = 0, i.e. no derivate).
%H A231864 E. J. Barbeau, <a href="http://cms.math.ca/cmb/v4/p117">Remark on an arithmetic derivative</a>, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.
%F A231864 a(n) = sum((i')^2, i=1..n) where i'=A003415.
%e A231864 (1')^2+(2')^2+(3')^2+(4')^2+(5')^2=0+1+1+16+1=19->a(5)=19.
%p A231864 der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^2,i=1..j),j=1..45);
%t A231864 dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Accumulate[Table[dn[n]^2, {n, 100}]] (* _T. D. Noe_, Nov 20 2013 *)
%Y A231864 Cf. A003415, A190121, A231946
%K A231864 nonn
%O A231864 1,3
%A A231864 _Giorgio Balzarotti_, Nov 14 2013