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%I #23 Jan 21 2023 02:22:29
%S 0,1,2,66,67,192,193,1921,2137,2480,2481,6577,6578,7307,7819,40587,
%T 40588,49849,49850,63674,64674,66871,66872,152056,153056,156431,
%U 176114,208882,208883,238674,238675,750675,753419,760278,762006,978006,978007,987268,991364
%N Partial sums of the third power of the arithmetic derivative function A003415.
%C a(n) grows roughly like 0.66*n^4 as n->oo.
%C Note: 1^3 + 2^3 + 3^3 + 4^3 + 5^3 ... -> ~ (1/4)*n^4; the asymptotic similarity between the sum of powers of positive integers and the sum of powers of their derivatives stands also with sums in which the terms are higher powers, i.e., Sum_{j=1..n} j'^m -> k*n^(m+1) as Sum_{j=i..n} j^m -> h*n^(m+1) when n->oo, in other words, the ratio of the two sums is a constant.
%H Robert Israel, <a href="/A231946/b231946.txt">Table of n, a(n) for n = 1..10000</a>
%H E. J. Barbeau, <a href="https://doi.org/10.4153/CMB-1961-013-0">Remark on an arithmetic derivative</a>, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.
%F a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).
%e (1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.
%p der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(i)^3,i=1..j),j=1..60);
%t 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]^3, {n, 100}]] (* _T. D. Noe_, Nov 20 2013 *)
%Y Cf. A003415, A190121, A231864.
%K nonn
%O 1,3
%A _Giorgio Balzarotti_, Nov 15 2013
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