%I A006880 M3608
%S A006880 0,4,25,168,1229,9592,78498,664579,5761455,50847534,455052511,4118054813,
%T A006880 37607912018,346065536839,3204941750802,29844570422669,279238341033925,
%U A006880 2623557157654233,24739954287740860,234057667276344607,2220819602560918840
%N A006880 Number of primes < 10^n.
%C A006880 Number of primes with at most n digits; or pi(10^n).
%C A006880 Also omega( (10^n)! ), where omega(x): number of distinct prime divisors 
               of x. - Cino Hilliard (hillcino368(AT)hotmail.com), Jul 04 2007
%C A006880 This sequence also gives a good approximation for the sum of primes < 
               10^(n/2). This is evident from the fact that the number of primes 
               < 10^2n closely approximates the sum of primes < 10^n. See link on 
               Sum of Primes for the derivation. - Cino Hilliard (hillcino368(AT)hotmail.com), 
               Jun 08 2008
%D A006880 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 
               Springer, NY, 2001; see p. 11.
%D A006880 A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 
               113 (No. 1, 2006), 1-33.
%D A006880 C. T. Long, Elementary Introduction to Number Theory. Prentice-Hall, 
               Englewood Cliffs, NJ, 1987, p. 77.
%D A006880 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 179.
%D A006880 H. Riesel, "Prime numbers and computer methods for factorization," Progress 
               in Mathematics, Vol. 57, Birkhauser, Boston, 1985, page 38.
%D A006880 M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 
               2003; see p. 48.
%D A006880 D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 
               2nd edition, 1978, p. 15.
%D A006880 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006880 N. J. A. Sloane, <a href="b006880.txt">Table of n, a(n) for n=0..23</
               a> (from the web page of Tomas Oliveira e Silva)
%H A006880 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml">
               How Many Primes Are There?</a>
%H A006880 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/notes/md.html">
               Mark Deleglise's work</a>
%H A006880 Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/
               Primes/pixtable.html">a(22) found by pi(x) project</a>
%H A006880 Xavier Gourdon & Pascal Sebah, <a href="http://numbers.computation.free.fr/
               Constants/Primes/Pix/results.html">The pi(x) project : results and 
               current computations</a>
%H A006880 A. Granville and G. Martin, <a href="http://www.arXiv.org/abs/math.NT/
               0408319">Prime number races</a>
%H A006880 Cino Hilliard, <a href="http://docs.google.com/Doc?docid=dgpq9w4b_26dtrq634m&hl=en"> 
               Sum of primes </a>
%H A006880 R. K. Hoeflin, <a href="http://www.eskimo.com/~miyaguch/titan.html">Titan 
               Test</a>
%H A006880 J. C. Lagarias, V. S. Miller and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/
               ~odlyzko/doc/cnt.html">Computing pi(x): The Meissel-Lehmer method</
               a>, Math. Comp., 44 (1985), pp. 537-560.
%H A006880 J. C. Lagarias and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/
               doc/cnt.html">Computing pi(x): An analytic method</a>, J. Algorithms, 
               8 (1987), pp. 173-191.
%H A006880 Tomas Oliveira e Silva, <a href="http://www.ieeta.pt/~tos/primes.html">
               Tables of values of pi(x) and of pi2(x)</a>
%H A006880 Tomas Oliveira e Silva, <a href="http://www.ieeta.pt/~tos/bib/5.4.pdf">
               Computing (x): the combinatorial method</a>, REVISTA DO DETUA, VOL. 
               4, N 6, MARCH 2006.
%H A006880 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/ss-a.htm">
               The distribution of prime numbers</a>
%H A006880 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeCountingFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A006880 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">
               Prime number theorem</a>
%H A006880 <a href="Sindx_Pri.html#primepop">Index entries for sequences related 
               to numbers of primes in various ranges</a>
%F A006880 Partial sums of A006879. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 
               25 2004
%t A006880 Table[PrimePi[10^n], {n, 0, 16}]
%o A006880 (PARI) g(n) = for(x=0,n,print1(omega((10^x)!),",")) - Cino Hilliard (hillcino368(AT)hotmail.com), 
               Jul 04 2007
%Y A006880 Cf. A000720, A006879, A007053, A040014, A006988.
%Y A006880 Sequence in context: A073517 A074422 A128419 this_sequence A081068 A163072 
               A140177
%Y A006880 Adjacent sequences: A006877 A006878 A006879 this_sequence A006881 A006882 
               A006883
%K A006880 nonn,hard,nice
%O A006880 0,2
%A A006880 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006880 Lehmer gave the incorrect value 455052512 for the 10th term. More terms 
               5/96. Jud McCranie (j.mccranie(AT)comcast.net) points out that the 
               11-th term is not 4188054813 but rather 4118054813.
%E A006880 a(22) from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 04 2001
%E A006880 a(23) (see Gourdon and Sebah) has yet to be verified and the assumed 
               error is +/-1. - Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 10 2002. 
               The actual error was 14037804. - N. J. A. Sloane (njas(AT)research.att.com), 
               Nov 28 2007
%E A006880 a(23) corrected by N. J. A. Sloane (njas(AT)research.att.com) from the 
               web page of Tomas Oliveira e Silva, Nov 28 2007