A025201 -id:A025201 - cp's OEIS frontend

A006880 Number of primes < 10^n.

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

Offset: 0

N. J. A. Sloane and Simon Plouffe

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

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 143, 146.
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. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
Paulo 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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 455052511 at p. 190.

David Baugh, Table of n, a(n) for n = 0..29 (terms n = 1..27 from Charles R Greathouse IV).
Chris K. Caldwell, How Many Primes Are There?
Chris K. Caldwell, Mark Deleglise's work
Muhammed Hüsrev Cilasun, An Analytical Approach to Exponent-Restricted 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.
Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, A practical analytic method for calculating pi(x), Math. Comp. 86 (2017), 2889-2909.
Xavier Gourdon, a(22) found by pi(x) project
Xavier Gourdon and 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), 1-33.
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
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979, [see p. 6 of the pdf].
Rishi Kumar, Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p, arXiv:2406.05890 [math.NT], 2024. See p. 7.
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 Andrew M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191.
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], 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.
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
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

Cf. A000720, A006879, A006988, A007053, A011557, A025201, A040014.

a006880 = a000720 . (10 ^)  -- Reinhard Zumkeller, Mar 17 2015

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 *)

a(n)=primepi(10^n) \\ Charles R Greathouse IV, Nov 08 2011

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

Limit_{n->oo} a(n)/a(n-1) = 10. - Stefano Spezia, Aug 31 2025

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
a(28) in the b-file from David Baugh and Kim Walisch, Oct 26 2020
a(29) in the b-file from David Baugh and Kim Walisch, Feb 28 2022

A046654 Nearest integer to Sum_{k=1..n} log(k) = log(n!).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20, 23, 25, 28, 31, 34, 36, 39, 42, 45, 48, 52, 55, 58, 61, 65, 68, 71, 75, 78, 82, 85, 89, 92, 96, 99, 103, 107, 110, 114, 118, 122, 125, 129, 133, 137, 141, 145, 148, 152, 156, 160, 164, 168, 172, 176, 180

Offset: 0

N. J. A. Sloane, Dec 27 1999

nonn

a(n) is also the nearest integer to log(n!). - Eric M. Schmidt, Jun 19 2015
Log(n!) is asymptotic to A275341. - Mats Granvik, Aug 02 2016
Stirling's approximation s(n) = n*log(n) - n + log(2*Pi*n)/2 is known to be equal to log(n!) up to an error between 1/(12n + 1) and 1/12n. For all 0 < n < 10^6 except for n = 11, round(s(n)) = a(n). What is the next such exceptional index n? - M. F. Hasler, Dec 03 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 22.1.

Eric M. Schmidt, Table of n, a(n) for n = 0..10000 (corrected by Sean A. Irvine, Jan 18 2019)

Cf. A025201.

[Round(Log(Factorial(n))): n in [2..100]]; // Vincenzo Librandi, Jun 19 2015

nn = 58; t = Accumulate[Log /@ Range[nn]]; Table[If[(y = Ceiling[x = t[[i]]]) - x <= x - (z = Floor[x]), a = y, a = z]; a, {i, nn}] (* Jayanta Basu, Jun 27 2013 *)

A046654(n)=round(lngamma(n+1)) \\ M. F. Hasler, Dec 03 2018

a(n) = n*log(n) - n + O(log(n)). - Arkadiusz Wesolowski, Oct 18 2013
a(n) = round(LogGamma(n + 1)). - Mats Granvik, Roger L. Bagula, Aug 06 2016
a(n) = round(log(Product_{k=1..n} A139547(n,k))). - Mats Granvik, Aug 07 2016

Name edited and a(0) = 0 prepended by M. F. Hasler, Dec 03 2018

A056690 a(n) = floor(product_{k=2..n} log(k)).

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 12, 27, 62, 149, 371, 952, 2513, 6807, 18873, 53472, 154554, 455076, 1363287, 4150560, 12829558, 40227005, 127843589, 411512640, 1340747909, 4418886387, 14724633149, 49582195801, 168638834537, 579103599979

Offset: 2

Leroy Quet, Aug 10 2000

			log(2)*log(3)*log(4)*log(5)= 1.699... So a(5) = 1, since the integer part of 1.699... is 1.

Vaclav Kotesovec, Table of n, a(n) for n = 2..1000

Cf. A025201, A046654.

Table[Floor[Product[Log[k], {k, 2, n}]], {n, 2, 40}] (* Vaclav Kotesovec, Nov 28 2015 *)

log(a(n)) ~ n*(log(log(n)) - 1/log(n) - 1/log(n)^2 - 2/log(n)^3 - 6/log(n)^4 - 24/log(n)^5 - ...). - Vaclav Kotesovec, Feb 12 2023

A066026 a(n) = ceiling(log(n!)).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 11, 13, 16, 18, 20, 23, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 59, 62, 65, 68, 72, 75, 79, 82, 86, 89, 93, 96, 100, 103, 107, 111, 115, 118, 122, 126, 130, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 202

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 11 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A025201, A050502, A034886.

Ceiling[Log[Range[70]!]] (* Harvey P. Dale, Jul 23 2012 *)

{ for (n=1, 1000, a=ceil(log(n!)); write("b066026.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 07 2009

a(n) = ceil(lngamma(n+1)) \\ Michel Marcus, Jun 29 2015

Terms a(51)-a(63) from Harry J. Smith, Nov 07 2009

A258782 Nearest integer to log_2(n!).

Original entry on oeis.org

0, 0, 1, 3, 5, 7, 9, 12, 15, 18, 22, 25, 29, 33, 36, 40, 44, 48, 53, 57, 61, 65, 70, 74, 79, 84, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 149, 154, 159, 165, 170, 175, 181, 186, 192, 197, 203, 209, 214, 220, 226, 231, 237, 243, 249, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314

Offset: 0

Eli Sadoff, Jun 10 2015

			a(6) = round(log_2(6!)) = round(9.49...) = 9.

Cf. A025201, A067850.

for i = 1:20 { disp(round(log2(factorial(i)))) } end

[Round(LogGamma(n+1)/Log(2)): n in [0..70]]; // Bruno Berselli, Jun 23 2015

seq(round(lnGAMMA(n+1)/ln(2)),n=0..100); # Robert Israel, Jun 10 2015

Round[Log[2, Range[0, 100]! ]] (* Giovanni Resta, Jun 10 2015 *)

a(n) = round(log(n!)/log(2)); \\ Michel Marcus, Jun 10 2015

a(n)=round(lngamma(n+1)/log(2)) \\ Charles R Greathouse IV, Jun 10 2015

[round(log_gamma(n+1)/log2) for n in (0..70)] # Bruno Berselli, Jun 23 2015

a(n) = round(log_2(n!)).
a(n) = A004257(A000142(n)). - Michel Marcus, Jun 10 2015
a(n) = round(Sum_{k=1..n} log_2(k)). - Tom Edgar, Jun 10 2015
a(n) is within 1 of n*(log(n)-1)/log(2) + log(n)/(2*log(2)) + log(sqrt(2*Pi))/log(2) for n >= 1. - Robert Israel, Jun 10 2015

A321991 a(n) = round(1/frac(log(n!))) for n > 1; a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 1, 6, 1, 2, 2, 2, 1, 10, 2, 1, 2, 5, 1, 1, 2, 3, 3, 3, 3, 2, 2, 1, 277, 4, 2, 1, 4, 2, 11, 2, 18, 2, 7, 1, 3, 1, 2, 3, 29, 1, 2, 3, 8, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 1, 1, 107, 6, 3, 2, 1, 1, 5, 2, 1, 1, 4, 2, 1, 5, 2, 1, 3, 1, 15, 2, 1, 3, 1, 5, 1, 6, 2, 7

Offset: 0

M. F. Hasler, Dec 03 2018

nonn

Motivated by Stirling's approximation s(n) = n*log(n) - n - log(2*Pi*n)/2 of log(n!), known to verify s(n) + 1/(12n+1) < log(n!) < s(n) + 1/12n. s(n) has the same integer part as log(n!) for all 1 < n < 10^6 at least, cf. A025201, but if the fractional part of log(n!) is less than 1/(12n+1), the approximation would yield the next lower integer. The first such n must have a(n) > 12n, so it is necessarily a record in this sequence, even a record of the sequence (a(n)/n). a(24) = 277 is a close miss, 12*24 = 288.

			Records occur at a(2) = 1, a(4) = 6, a(10) = 10, a(25) = 277, a(589) = 760, a(2965) = 921, a(3295) = 988, a(3802) = 1326, a(8743) = 1516, a(10634) = 2458, a(15404) = 11472, a(31672) = 56377, a(152170) = 162958, a(307001) = 295209, a(704236) = 491928, a(862929) = 528736, a(904492) = 1612903, a(1356678) = 5098244, ...
Among these, only a(2) = 1, a(4) = 6, and a(25) = 277 set a record for a(n)/n. No value is known for which a(n)/n >= 12.

Cf. A025201, A046654.

A321991(n)=if(n>1,1\/frac(lngamma(n+1)),0)

cp's OEIS Frontend

A006880 Number of primes < 10^n.

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A046654 Nearest integer to Sum_{k=1..n} log(k) = log(n!).

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A056690 a(n) = floor(product_{k=2..n} log(k)).

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A066026 a(n) = ceiling(log(n!)).

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A258782 Nearest integer to log_2(n!).

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A321991 a(n) = round(1/frac(log(n!))) for n > 1; a(0) = a(1) = 0.

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A025209 a(n) = floor(floor(S2)/floor(S1)), where S2 and S1 are, respectively, the 2nd and first elementary symmetric functions of {log(k)}, k = 2,3,...,n.

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