OFFSET
1,3
COMMENTS
a(n)-> ~ 0.4*n^3 as n-> oo (note: 1^2+2^2+3^3+4^4+5^4 ...-> ~ 1/3*n^3)
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).
LINKS
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.
FORMULA
a(n) = sum((i')^2, i=1..n) where i'=A003415.
EXAMPLE
(1')^2+(2')^2+(3')^2+(4')^2+(5')^2=0+1+1+16+1=19->a(5)=19.
MAPLE
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);
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]^2, {n, 100}]] (* T. D. Noe, Nov 20 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, Nov 14 2013
STATUS
approved