A190121 - OEIS

%I #38 Oct 06 2023 10:53:24

%S 0,1,2,6,7,12,13,25,31,38,39,55,56,65,73,105,106,127,128,152,162,175,

%T 176,220,230,245,272,304,305,336,337,417,431,450,462,522,523,544,560,

%U 628,629,670,671,719,758,783,784,896,910,955,975,1031,1032,1113,1129

%N Partial sums of the arithmetic derivative function A003415.

%C See A229523 for a(10^n). - _M. F. Hasler_, Sep 25 2013

%H G. C. Greubel, <a href="/A190121/b190121.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)

%H E. J. Barbeau, <a href="https://doi.org/10.4153/CMB-1961-013-0">Remark on an arithmetic derivative</a>, Canad. Math. Bull., Vol. 4, No. 2 (May 1961), pp. 117-122.

%F 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

%F 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

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

%p der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):

%p seq(add(der(i),i=1..j),j=1..100);

%t 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 *)

%t 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 *)

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

%o (PARI) A190121(n)=sum(k=1,n,A003415(k)) \\ _M. F. Hasler_, Sep 26 2013

%Y Cf. A003415, A136141, A229523.

%K nonn,easy

%O 1,3

%A _Giorgio Balzarotti_, May 04 2011