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A190121 Partial sums of the arithmetic derivative function A003415. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.

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-Fran├žois Alcover, Feb 21 2014 *)

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.

Sequence in context: A217575 A172154 A072147 * A105329 A124319 A078471

Adjacent sequences:  A190118 A190119 A190120 * A190122 A190123 A190124

KEYWORD

nonn

AUTHOR

Giorgio Balzarotti, May 04 2011

STATUS

approved

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Last modified November 23 00:45 EST 2014. Contains 249836 sequences.