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 A006880 Number of primes < 10^n. (Formerly M3608) 198
 0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290, 1925320391606803968923, 18435599767349200867866, 176846309399143769411680, 1699246750872437141327603 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of primes with at most n digits; or pi(10^n). Partial sums of A006879. - Lekraj Beedassy, Jun 25 2004 Also omega( (10^n)! ), where omega(x): number of distinct prime divisors of x. - Cino Hilliard, Jul 04 2007 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, Jun 08 2008 (See A025201) approximately it appears that (10^n)/log((n+3)!) < A006880. - Eric Desbiaux, Jul 20 2010 REFERENCES R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11. M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 48. C. T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77. P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179. H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, page 38. D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 2nd edition, 1978, p. 15. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Charles R Greathouse IV, Table of n, a(n) for n = 0..27 J. Buethe, J. Franke, A. Jost, and T. Kleinjung, "Conditional Calculation of pi(10^24)", Posting to the Number Theory Mailing List, Jul 29 2010. [archived copy] C. K. Caldwell, How Many Primes Are There? C. K. Caldwell, Mark Deleglise's work M. H. Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014. M. H. Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3. Xavier Gourdon, a(22) found by pi(x) project Xavier Gourdon & Pascal Sebah, The pi(x) project : results and current computations A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33. A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004. Cino Hilliard, Sum of primes [unusable link] R. K. Hoeflin, Titan Test D. S. Kluk and N. J. A. Sloane, Correspondence, 1979, [see p. 6 of the pdf] J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560. J. C. Lagarias and A. M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191. G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy) Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do Detua, Vol. 4, N 6, March 2006. David J. Platt, Computing pi(x) analytically, arXiv:1203.5712 [math.NT], 2012-2013. Vladimir Pletser, Conjecture on the value of Pi(10^26), the number of primes less than 10^26 arXiv:1307.4444 [math.NT], 2013. Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015. M. R. Watkins, The distribution of prime numbers Eric Weisstein's World of Mathematics, Prime Counting Function Wikipedia, Prime number theorem R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989 MATHEMATICA Table[PrimePi[10^n], {n, 0, 16}] (* Second program (Mma [V.11] can't compute for n>14): *) Unprotect[PrimePi]; PrimePi[10^13] = 346065536839; PrimePi[10^14] = 3204941750802; PrimePi[10^15] = 29844570422669; PrimePi[10^16] = 279238341033925; PrimePi[10^17] = 2623557157654233; PrimePi[10^18] = 24739954287740860; PrimePi[10^19] = 234057667276344607; PrimePi[10^20] = 2220819602560918840; PrimePi[10^21] = 21127269486018731928; PrimePi[10^22] = 201467286689315906290; PrimePi[10^23] = 1925320391606803968923; PrimePi[10^24] = 18435599767349200867866; PrimePi[10^25] = 176846309399143769411680; PrimePi[10^26] = 1699246750872437141327603; PrimePi[10^27] = 16352460426841680446427399; Table[PrimePi[10^n], {n, 0, 27}] (* Jean-François Alcover, Nov 08 2016, from latest b-file *) PROG (PARI) g(n) = for(x=0, n, print1(omega((10^x)!), ", ")) \\ Cino Hilliard, Jul 04 2007 (PARI) a(n)=primepi(10^n) \\ Charles R Greathouse IV, Nov 08 2011 (Haskell) a006880 = a000720 . (10 ^)  -- Reinhard Zumkeller, Mar 17 2015 CROSSREFS Cf. A000720, A006879, A007053, A040014, A006988, A011557. Sequence in context: A226945 A225137 A229255 * A227693 A175255 A081068 Adjacent sequences:  A006877 A006878 A006879 * A006881 A006882 A006883 KEYWORD nonn,hard,nice AUTHOR EXTENSIONS Lehmer gave the incorrect value 455052512 for the 10th term. More terms May 1996. Jud McCranie points out that the 11th term is not 4188054813 but rather 4118054813. a(22) from Robert G. Wilson v, Sep 04 2001 a(23) (see Gourdon and Sebah) has yet to be verified and the assumed error is +-1. - Robert G. Wilson v, Jul 10 2002. The actual error was 14037804. - N. J. A. Sloane, Nov 28 2007 a(23) corrected by N. J. A. Sloane from the web page of Tomás Oliveira e Silva, Nov 28 2007 a(25) from J. Buethe, J. Franke, A. Jost, T. Kleinjung, Jun 01 2013, who said: "We have calculated pi(10^25)=176846309399143769411680 unconditionally, using an analytic method based on Weil's explicit formula". a(26) from Douglas B. Staple, Dec 02 2014 a(27) in the b-file from David Baugh and Kim Walisch via Charles R Greathouse IV, Jun 01 2016 STATUS approved

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