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%I A229523 #43 Oct 06 2023 10:53:38
%S A229523 0,38,3906,386517,38671110,3865941752,386580463478,38657862140521,
%T A229523 3865783461518530,386578337105347684,38657833484501788407,
%U A229523 3865783345588492717623,386578334529872234861944,38657833452536035472588254,3865783345249467526546175599
%N A229523 Partial sum of the arithmetic derivative A003415 (A190121) up to 10^n.
%H A229523 Hiroaki Yamanouchi, <a href="/A229523/b229523.txt">Table of n, a(n) for n = 0..17</a>
%H A229523 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 A229523 a(n) = A190121(10^n).
%F A229523 It seems that a(n)/10^(2n-1) -> 3.865783... as n -> oo.
%F A229523 Note: A190121 ~ 0.374... * n^2 [Barbeau]. - _Giorgio Balzarotti_, Oct 15 2013
%F A229523 a(n) ~ 0.386578334524897563932183729927 * 100^n. - _Hiroaki Yamanouchi_, Jul 09 2014
%F A229523 The constant is (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
%o A229523 (PARI) s=0;for(k=0,8,for(n=10^(k-1)+1,10^k,s+=A003415(n));print1(s","));s
%Y A229523 Cf. A003415, A136141, A190121.
%K A229523 nonn,hard
%O A229523 0,2
%A A229523 _M. F. Hasler_, Sep 25 2013
%E A229523 a(8)-a(10) from _Donovan Johnson_, Sep 25 2013
%E A229523 a(11)-a(12) from _Giovanni Resta_, Mar 13 2014
%E A229523 a(13)-a(14) from _Hiroaki Yamanouchi_, Jul 09 2014

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