cp's OEIS Frontend

A231946 Partial sums of the third power of the arithmetic derivative function A003415.

%I A231946 #23 Jan 21 2023 02:22:29
%S A231946 0,1,2,66,67,192,193,1921,2137,2480,2481,6577,6578,7307,7819,40587,
%T A231946 40588,49849,49850,63674,64674,66871,66872,152056,153056,156431,
%U A231946 176114,208882,208883,238674,238675,750675,753419,760278,762006,978006,978007,987268,991364
%N A231946 Partial sums of the third power of the arithmetic derivative function A003415.
%C A231946 a(n) grows roughly like 0.66*n^4 as n->oo.
%C A231946 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 A231946 Robert Israel, <a href="/A231946/b231946.txt">Table of n, a(n) for n = 1..10000</a>
%H A231946 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 A231946 a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).
%e A231946 (1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.
%p A231946 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 A231946 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 A231946 Cf. A003415, A190121, A231864.
%K A231946 nonn
%O A231946 1,3
%A A231946 _Giorgio Balzarotti_, Nov 15 2013