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

0, 1, 2, 18, 19, 44, 45, 189, 225, 274, 275, 531, 532, 613, 677, 1701, 1702, 2143, 2144, 2720, 2820, 2989, 2990, 4926, 5026, 5251, 5980, 7004, 7005, 7966, 7967, 14367, 14563, 14924, 15068, 18668, 18669, 19110, 19366, 23990, 23991, 25672, 25673, 27977, 29498

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

Table of n, a(n) for n=1..45.

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

Cf. A003415, A190121, A231946

Sequence in context: A083502 A076378 A231502 * A066242 A022371 A299380

Adjacent sequences: A231861 A231862 A231863 * A231865 A231866 A231867

Keyword

nonn

Author

Giorgio Balzarotti, Nov 14 2013

Status

approved