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%I #66 May 11 2022 10:37:02
%S 2,5,10,17,38,130,339,754,1701,3104,11588,38263,108971,314890,1052619,
%T 3214632,7956589,21949555,99877775,222744644,597394254,1932355208,
%U 7250186216,17146907278,55160980939,155891678121,508666658006,1427745660374,4551193622464
%N Difference between nearest integer to Li(10^n) and pi(10^n), where Li(x) = integral of log(x) and pi(10^n) = number of primes <= 10^n (A006880).
%C On his prime pages C. K. Caldwell remarks: "However in 1914 Littlewood proved that pi(x)-Li(x) assumes both positive and negative values infinitely often". - _Frank Ellermann_, May 31 2003
%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1995, page 146.
%D Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see table on p. 90.
%H Chris K. Caldwell, How many primes are there, table, <a href="http://www.utm.edu/research/primes/howmany.shtml#table">Values of pi(x)</a>.
%H Chris K. Caldwell, How many primes are there, table, <a href="http://www.utm.edu/research/primes/howmany.shtml#table3">Approximations to pi(x)</a>.
%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html">Counting the primes</a>
%H Andrew Granville, <a href="http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf">Harald Cramer and the Distribution of Prime Numbers</a>
%H Anatolii A. Karatsuba and Ekatherina A. Karatsuba, <a href="https://web.archive.org/web/20171109081433/http://www.phy.bg.ac.rs/~mphys6/proceedings5/14-KaratsubaKaratsuba.ps">The "problem of remainders" in theoretical physics: "physical zeta" function</a>, 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 14-23 September 2010, Belgrade, Serbia. [From Internet Archive Wayback Machine]
%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/primes.html">Tables of values of pi(x) and of pi2(x)</a>
%H Y. Saouter and P. Demichel, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02351-3">A sharp region where pi(x)-li(x) is positive</a>, Math. Comp. 79 (272) (2010) 2395-2405. [From _R. J. Mathar_, Oct 08 2010]
%H Munibah Tahir, <a href="http://eprints.ma.man.ac.uk/1541/01/Munibah2010.pdf">A new bound for the smallest x with pi(x) > li(x)</a> (2010).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>
%t Table[Round[LogIntegral[10^n] - PrimePi[10^n]], {n, 1, 13}]
%o (PARI) A057752=vector(#A006880,i,round(-eint1(-log(10^i))-A006880[i])) \\ _M. F. Hasler_, Feb 26 2008
%o (Python)
%o from sympy import N, li, primepi, floor
%o def round(n):
%o return int(floor(n+0.5))
%o def A057752(n):
%o return round(N(li(10**n),10*n)) - primepi(10**n) # _Chai Wah Wu_, Apr 30 2018
%Y Cf. A006880, A052435, A057794.
%K sign,hard
%O 1,1
%A _Robert G. Wilson v_, Oct 30 2000
%E More terms from _Frank Ellermann_, May 31 2003
%E The value of a(23) is not known at present, I believe. - _N. J. A. Sloane_, Mar 17 2008
%E Name corrected and extended for last two terms a(23) and a(24), with Pi(10^n) for n=23 and 24 from A006880, by _Vladimir Pletser_, Mar 10 2013
%E a(25)-a(27) added, using data from A006880, by _Chai Wah Wu_, Apr 30 2018
%E a(28) added, using data from A006880, by _Eduard Roure Perdices_, Apr 14 2021
%E a(29) added, using data from A006880, by _Reza K Ghazi_, May 10 2022
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