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

0, 1, 2, 66, 67, 192, 193, 1921, 2137, 2480, 2481, 6577, 6578, 7307, 7819, 40587, 40588, 49849, 49850, 63674, 64674, 66871, 66872, 152056, 153056, 156431, 176114, 208882, 208883, 238674, 238675, 750675, 753419, 760278, 762006, 978006, 978007, 987268, 991364

Offset: 1

Author

Giorgio Balzarotti, Nov 15 2013

Keywords

• nonn

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.

Examples

			(1')^3 + (2')^3 + (3')^3 + (4')^3 + (5')^3 = 0+1+1+64+1 = 67, so a(5)=67.
		

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.

Crossrefs

Cf. A003415, A190121, A231864.

Programs

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 *)

Formula

a(n) = Sum_{j=1..n} (j')^3, where j' = A003415(j).