

A006880


Number of primes < 10^n.
(Formerly M3608)


222



0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290
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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 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
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


REFERENCES

Richard Crandall and Carl B. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; p. 11.
Keith Devlin, Mathematics: The New Golden Age, new and revised edition. New York: Columbia University Press (1993): p. 6, Table 1.
Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; p. 48.
Calvin T. Long, Elementary Introduction to Number Theory. PrenticeHall, Englewood Cliffs, NJ, 1987, p. 77.
Paulo Ribenboim, The Book of Prime Number Records. SpringerVerlag, 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

David Baugh, Table of n, a(n) for n = 0..29 (terms n = 1..27 from Charles R Greathouse IV).
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]
Chris K. Caldwell, How Many Primes Are There?
Chris K. Caldwell, Mark Deleglise's work
Muhammed Hüsrev Cilasun, An Analytical Approach to ExponentRestricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
Muhammed Hüsrev 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
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 133.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Cino Hilliard, Sum of primes [unusable link]
Ronald K. Hoeflin, Titan Test
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the BatemanHorn conjecture, 2021.
Gareth A. Jones and Alexander K. Zvonkin, Orders of simple groups and the BatemanHorn Conjecture, arXiv:2209.06510 [math.GR], 2022.
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 MeisselLehmer method, Math. Comp., 44 (1985), pp. 537560.
J. C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173191.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 7999. (Annotated scanned copy)
Pieter Moree, Izabela Petrykiewicz, and Alisa Sedunova, A computational history of prime numbers and Riemann zeros, arXiv:1810.05244 [math.NT], 2018. See Table 1 p. 6.
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], 20122013.
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
Robert G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences related to numbers of primes in various ranges


FORMULA

a(n) = A000720(10^n).  M. F. Hasler, Dec 03 2018


MATHEMATICA

Table[PrimePi[10^n], {n, 0, 14}] (* JeanFrançois Alcover, Nov 08 2016, corrected Sep 29 2020, a(14) being the maximum computable with certain implementations *)


PROG

(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,changed


AUTHOR

N. J. A. Sloane and Simon Plouffe


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 bfile from David Baugh and Kim Walisch via Charles R Greathouse IV, Jun 01 2016
a(28) in the bfile from David Baugh and Kim Walisch, Oct 26 2020
a(29) in the bfile from David Baugh and Kim Walisch, Feb 28 2022


STATUS

approved



