A229523 Partial sum of the arithmetic derivative A003415 (A190121) up to 10^n.
0, 38, 3906, 386517, 38671110, 3865941752, 386580463478, 38657862140521, 3865783461518530, 386578337105347684, 38657833484501788407, 3865783345588492717623, 386578334529872234861944, 38657833452536035472588254, 3865783345249467526546175599
Offset: 0
Links
- Hiroaki Yamanouchi, Table of n, a(n) for n = 0..17
- E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., Vol. 4, No. 2 (May 1961), pp. 117-122.
Programs
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PARI
s=0;for(k=0,8,for(n=10^(k-1)+1,10^k,s+=A003415(n));print1(s","));s
Formula
a(n) = A190121(10^n).
It seems that a(n)/10^(2n-1) -> 3.865783... as n -> oo.
Note: A190121 ~ 0.374... * n^2 [Barbeau]. - Giorgio Balzarotti, Oct 15 2013
a(n) ~ 0.386578334524897563932183729927 * 100^n. - Hiroaki Yamanouchi, Jul 09 2014
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
Extensions
a(8)-a(10) from Donovan Johnson, Sep 25 2013
a(11)-a(12) from Giovanni Resta, Mar 13 2014
a(13)-a(14) from Hiroaki Yamanouchi, Jul 09 2014