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A006880 Number of primes < 10^n.
(Formerly M3608)
196
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).

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

M. H. Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.

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.

A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

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

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, 2014

Xavier Gourdon, a(22) found by pi(x) project

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]

Xavier Gourdon & Pascal Sebah, The pi(x) project : results and current computations

A. Granville and G. Martin, Prime number races

Cino Hilliard, Sum of primes

R. K. Hoeflin, Titan Test

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.

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 (2012)

Vladimir Pletser, Conjecture on the value of Pi(10^26), the number of primes less than 10^26 (2013)

Douglas B. Staple, The combinatorial algorithm for computing pi(x), preprint, 2015.

M. R. Watkins, The distribution of prime numbers

Eric Weisstein's World of Mathematics, Prime Counting Function

Wikipedia, Prime number theorem

Index entries for sequences related to numbers of primes in various ranges

FORMULA

Partial sums of A006879. - Lekraj Beedassy, Jun 25 2004

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,changed

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Lehmer gave the incorrect value 455052512 for the 10th term. More terms 5/96. 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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Last modified December 10 21:13 EST 2016. Contains 279011 sequences.