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

0,2

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.

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

Prog

(PARI) s=0; for(k=0, 8, for(n=10^(k-1)+1, 10^k, s+=A003415(n)); print1(s", ")); s

Crossrefs

Cf. A003415, A136141, A190121.

Sequence in context: A221111 A322331 A282962 * A173133 A221354 A096558

Adjacent sequences: A229520 A229521 A229522 * A229524 A229525 A229526

Keyword

nonn,hard

Author

M. F. Hasler, Sep 25 2013

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

Status

approved