OFFSET
1,3
COMMENTS
a(n) grows roughly like 0.66*n^4 as n->oo.
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.
FORMULA
a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).
EXAMPLE
(1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, Nov 15 2013
STATUS
approved