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%I A006880 M3608 #216 Sep 18 2024 15:46:37
%S A006880 0,4,25,168,1229,9592,78498,664579,5761455,50847534,455052511,
%T A006880 4118054813,37607912018,346065536839,3204941750802,29844570422669,
%U A006880 279238341033925,2623557157654233,24739954287740860,234057667276344607,2220819602560918840,21127269486018731928,201467286689315906290
%N A006880 Number of primes < 10^n.
%C A006880 Number of primes with at most n digits; or pi(10^n).
%C A006880 Partial sums of A006879. - _Lekraj Beedassy_, Jun 25 2004
%C A006880 Also omega( (10^n)! ), where omega(x): number of distinct prime divisors of x. - _Cino Hilliard_, Jul 04 2007
%C A006880 This sequence also gives a good approximation for the sum of primes less than 10^(n/2). This is evident from the fact that the number of primes less than 10^2n closely approximates the sum of primes less than 10^n. See link on Sum of Primes for the derivation. - _Cino Hilliard_, Jun 08 2008
%C A006880 It appears that (10^n)/log((n+3)!) is a lower bound close to a(n), see A025201. - _Eric Desbiaux_, Jul 20 2010, edited by _M. F. Hasler_, Dec 03 2018
%D A006880 Richard Crandall and Carl B. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; p. 11.
%D A006880 Keith Devlin, Mathematics: The New Golden Age, new and revised edition. New York: Columbia University Press (1993): p. 6, Table 1.
%D A006880 Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; p. 48.
%D A006880 Calvin T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
%D A006880 Paulo 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 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).
%D A006880 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 455052511 at p. 190.
%H A006880 David Baugh, <a href="/A006880/b006880.txt">Table of n, a(n) for n = 0..29</a> (terms n = 1..27 from Charles R Greathouse IV).
%H A006880 Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml">How Many Primes Are There?</a>
%H A006880 Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/notes/md.html">Mark Deleglise's work</a>
%H A006880 Muhammed Hüsrev Cilasun, <a href="http://arxiv.org/abs/1412.3265">An Analytical Approach to Exponent-Restricted Multiple Counting Sequences</a>, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
%H A006880 Muhammed Hüsrev Cilasun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Cilasun/cila5.html">Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
%H A006880 Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, <a href="https://dx.doi.org/10.1090/mcom/3038">A practical analytic method for calculating pi(x)</a>, Math. Comp. 86 (2017), 2889-2909.
%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 and 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 Andrew Granville and Greg Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
%H A006880 Andrew Granville and Greg Martin, <a href="http://www.arXiv.org/abs/math.NT/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%H A006880 Cino Hilliard, <a href="http://docs.google.com/Doc?docid=dgpq9w4b_26dtrq634m&amp;hl=en">Sum of primes</a> [unusable link]
%H A006880 Ronald K. Hoeflin, <a href="http://miyaguchi.4sigma.org/hoeflin/titan/titan.html">Titan Test</a>
%H A006880 D. S. Kluk and N. J. A. Sloane, <a href="/A002050/a002050_3.pdf">Correspondence, 1979</a>, [see p. 6 of the pdf].
%H A006880 Rishi Kumar, <a href="https://arxiv.org/abs/2406.05890">Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p</a>, arXiv:2406.05890 [math.NT], 2024. See p. 7.
%H A006880 J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, <a href="https://doi.org/10.1090/S0025-5718-1985-0777285-5">Computing pi(x): The Meissel-Lehmer method</a>, Math. Comp., 44 (1985), pp. 537-560.
%H A006880 J. C. Lagarias and Andrew M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/arch/analytic.pi.of.x.pdf">Computing pi(x): An analytic method</a>, J. Algorithms, 8 (1987), pp. 173-191.
%H A006880 Pieter Moree, Izabela Petrykiewicz, and Alisa Sedunova, <a href="https://arxiv.org/abs/1810.05244">A computational history of prime numbers and Riemann zeros</a>, arXiv:1810.05244 [math.NT], 2018. See Table 1 p. 6.
%H A006880 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 A006880 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/bib/5.4.pdf">Computing pi(x): the combinatorial method</a>, Revista do Detua, Vol. 4, N 6, March 2006.
%H A006880 David J. Platt, <a href="http://arxiv.org/abs/1203.5712">Computing pi(x) analytically</a>, arXiv:1203.5712 [math.NT], 2012-2013.
%H A006880 Vladimir Pletser, <a href="http://arxiv.org/abs/1307.4444">Conjecture on the value of Pi(10^26), the number of primes less than 10^26</a>, arXiv:1307.4444 [math.NT], 2013.
%H A006880 Vladimir Pletser, <a href="https://doi.org/10.20944/preprints202402.0545.v1">Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems</a>, Preprints.org, 2024. See p. 20.
%H A006880 Douglas B. Staple, <a href="http://arxiv.org/abs/1503.01839">The combinatorial algorithm for computing pi(x)</a>, arXiv:1503.01839 [math.NT], 2015.
%H A006880 M. R. Watkins, <a href="https://web.archive.org/web/20060611061124/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">Prime Counting Function</a>
%H A006880 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>
%H A006880 Robert G. Wilson v, <a href="/A002982/a002982.pdf">Letter to N. J. A. Sloane, Jan. 1989</a>
%H A006880 <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>
%F A006880 a(n) = A000720(10^n). - _M. F. Hasler_, Dec 03 2018
%t A006880 Table[PrimePi[10^n], {n, 0, 14}] (* _Jean-François Alcover_, Nov 08 2016, corrected Sep 29 2020, a(14) being the maximum computable with certain implementations *)
%o A006880 (PARI) a(n)=primepi(10^n) \\ _Charles R Greathouse IV_, Nov 08 2011
%o A006880 (Haskell) a006880 = a000720 . (10 ^)  -- _Reinhard Zumkeller_, Mar 17 2015
%Y A006880 Cf. A000720, A006879, A006988, A007053, A011557, A025201, A040014.
%K A006880 nonn,hard,nice
%O A006880 0,2
%A A006880 _N. J. A. Sloane_ and _Simon Plouffe_
%E A006880 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.
%E A006880 a(22) from _Robert G. Wilson v_, 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_, Jul 10 2002 [The actual error was 14037804. - _N. J. A. Sloane_, Nov 28 2007]
%E A006880 a(23) corrected by _N. J. A. Sloane_ from the web page of Tomás Oliveira e Silva, Nov 28 2007
%E A006880 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".
%E A006880 a(26) from _Douglas B. Staple_, Dec 02 2014
%E A006880 a(27) in the b-file from _David Baugh_ and Kim Walisch via _Charles R Greathouse IV_, Jun 01 2016
%E A006880 a(28) in the b-file from _David Baugh_ and Kim Walisch, Oct 26 2020
%E A006880 a(29) in the b-file from _David Baugh_ and Kim Walisch, Feb 28 2022

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