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%I #36 Jan 25 2022 08:46:09
%S -1,4,3,4,2,-4,45,561,6549,69985,690493,6545056,60615397,555560046,
%T 5070271362,46223804313,421692578206,3853431791690,35289854434775,
%U 323979090116197,2981921009910364,27516571651291205,254562416350667928
%N Rounded difference between 10^n/(log(10^n) - A) and pi(10^n), where A is Legendre's constant and pi is the prime counting function.
%C Legendre's constant is 1.08366 (A228211). - _Alonso del Arte_, Nov 02 2013
%C This sequence has historical rather than mathematical interest, cf. A228211. It is better to use 1 + 1/log(10^n) instead of A. Since A is given to only 5 decimal places, it does not make much sense to compute terms of this sequence beyond n ~ 10. For n = 9, the error a(9)/A006880(9) is about 0.14%, while the error for 1 + 1/log(10^9) instead of A is only about 0.04%. - _M. F. Hasler_, Dec 03 2018
%D Jan Gullberg, "Mathematics, From the Birth of Numbers", W. W. Norton and Company, NY and London, 1997, page 81.
%H Eduard Roure Perdices, <a href="/A058290/b058290.txt">Table of n, a(n) for n = 0..28</a> (terms 0..23 from Harry J. Smith).
%F a(n) = round(10^n/(log(10^n) - 1.08366)) - A006880(n). - _M. F. Hasler_, Dec 03 2018
%t Table[ Round[ 10^n /(Log[10^n] - 1.08366) - PrimePi[10^n] ], {n, 0, 13} ]
%o (PARI) {A006880_vec = [0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290, 1925320391606803968923]} \\ Edited by _M. F. Hasler_, Dec 03 2018
%o {default(realprecision, 100); t=log(10); for (n=0, 23, write("b058290.txt", n, " ", round(10^n/(n*t - 1.08366)) - A006880_vec[n+1])))} \\ _Harry J. Smith_, Jun 22 2009
%o (PARI) A058290(n)={10^n\/(n*log(10)-1.08366)-A006880(n)} \\ with A006880(n)=primepi(10^n) and/or precomputed values for n > 10. - _M. F. Hasler_, Dec 03 2018
%Y Cf. A006880, A057752, A057794, A057835, A058289.
%K sign,less
%O 0,2
%A _Robert G. Wilson v_, Dec 07 2000
%E More terms from _Harry J. Smith_, Jun 22 2009
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