A190121 Partial sums of the arithmetic derivative function A003415.

0, 1, 2, 6, 7, 12, 13, 25, 31, 38, 39, 55, 56, 65, 73, 105, 106, 127, 128, 152, 162, 175, 176, 220, 230, 245, 272, 304, 305, 336, 337, 417, 431, 450, 462, 522, 523, 544, 560, 628, 629, 670, 671, 719, 758, 783, 784, 896, 910, 955, 975, 1031, 1032, 1113, 1129

Offset: 1

Author

Giorgio Balzarotti, May 04 2011

Keywords

• nonn
• easy

Comments

See A229523 for a(10^n). - M. F. Hasler, Sep 25 2013

Examples

			1'+2'+3'+4'+5' = 0+1+1+4+1 = 7 -> a(5) = 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
• E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., Vol. 4, No. 2 (May 1961), pp. 117-122.

Crossrefs

Cf. A003415, A136141, A229523.

Programs

Maple

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(der(i),i=1..j),j=1..100);

Mathematica

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; Table[d[n], {n, 1, 55}] // Accumulate (* Jean-François Alcover, Feb 21 2014 *)
A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]]; Table[Sum[A003415[k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Dec 29 2017 *)

PARI

s=0; A190121=vector(199,n,s+=A003415(n))

PARI

A190121(n)=sum(k=1,n,A003415(k)) \\ M. F. Hasler, Sep 26 2013

Formula

a(n)-> ~ 0.374*n^2 as n-> oo [Barbeau] (note: 1+2+3+4+5 ...-> ~ 1/2*n^2; the similarity stands also for higher power of the terms of sum). - Giorgio Balzarotti, Nov 14 2013

a(n) ~ c * n^2, where c = (1/2) * Sum_{p prime} 1/(p*(p-1)) = A136141 / 2 = 0.3865783345... . This constant was given by Barbeau (1961) but with the wrong value 0.374. - Amiram Eldar, Oct 06 2023