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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Giorgio Balzarotti, May 04 2011
STATUS
approved