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

Author

Giorgio Balzarotti, Nov 14 2013

Keywords

• nonn

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).

Examples

			(1')^2+(2')^2+(3')^2+(4')^2+(5')^2=0+1+1+16+1=19->a(5)=19.
		

Links

• E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., vol. 4, no. 2, May 1961, pp. 117-122.

Crossrefs

Cf. A003415, A190121, A231946

Programs

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 *)

Formula

a(n) = sum((i')^2, i=1..n) where i'=A003415.