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

1,3

Comments

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

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A003415, A136141, A229523.

Sequence in context: A172154 A293531 A072147 * A105329 A124319 A325262

Adjacent sequences: A190118 A190119 A190120 * A190122 A190123 A190124

Keyword

nonn,easy

Author

Giorgio Balzarotti, May 04 2011

Status

approved