A006879 -id:A006879 - cp's OEIS frontend

A000040 The prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

N. J. A. Sloane

See A065091 for comments, formulas etc. concerning only odd primes. For all information concerning prime powers, see A000961. For contributions concerning "almost primes" see A002808.
A number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p.
A natural number is prime if and only if it has exactly two (positive) divisors.
A prime has exactly one proper positive divisor, 1.
The paper by Kaoru Motose starts as follows: "Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. Then p is the order of 2 (mod q). Thus p is a divisor of q - 1 and q > p. This shows that there exist infinitely many prime numbers." - Pieter Moree, Oct 14 2004
1 is not a prime, for if the primes included 1, then the factorization of a natural number n into a product of primes would not be unique, since n = n*1.
Prime(n) and pi(n) are inverse functions: A000720(a(n)) = n and a(n) is the least number m such that a(A000720(m)) = a(n). a(A000720(n)) = n if (and only if) n is prime.
Second sequence ever computed by electronic computer, on EDSAC, May 09 1949 (see Renwick link). - Russ Cox, Apr 20 2006
Every prime p > 3 is a linear combination of previous primes prime(n) with nonzero coefficients c(n) and |c(n)| < prime(n). - Amarnath Murthy, Franklin T. Adams-Watters and Joshua Zucker, May 17 2006; clarified by Chayim Lowen, Jul 17 2015
The Greek transliteration of 'Prime Number' is 'Protos Arithmos'. - Daniel Forgues, May 08 2009 [Edited by Petros Hadjicostas, Nov 18 2019]
A number n is prime if and only if it is different from zero and different from a unit and each multiple of n decomposes into factors such that n divides at least one of the factors. This applies equally to the integers (where a prime has exactly four divisors (the definition of divisors is relaxed such that they can be negative)) and the positive integers (where a prime has exactly two distinct divisors). - Peter Luschny, Oct 09 2012
Motivated by his conjecture on representations of integers by alternating sums of consecutive primes, for any positive integer n, Zhi-Wei Sun conjectured that the polynomial P_n(x) = Sum_{k=0..n} a(k+1)*x^k is irreducible over the field of rational numbers with the Galois group S_n, and moreover P_n(x) is irreducible mod a(m) for some m <= n(n+1)/2. It seems that no known criterion on irreducibility of polynomials implies this conjecture. - Zhi-Wei Sun, Mar 23 2013
Questions on a(2n) and Ramanujan primes are in A233739. - Jonathan Sondow, Dec 16 2013
From Hieronymus Fischer, Apr 02 2014: (Start)
Natural numbers such that there is exactly one base b such that the base-b alternate digital sum is 0 (see A239707).
Equivalently: Numbers p > 1 such that b = p-1 is the only base >= 1 for which the base-b alternate digital sum is 0.
Equivalently: Numbers p > 1 such that the base-b alternate digital sum is <> 0 for all bases 1 <= b < p-1. (End)
An integer n > 1 is a prime if and only if it is not the sum of positive integers in arithmetic progression with common difference 2. - Jean-Christophe Hervé, Jun 01 2014
Conjecture: Numbers having prime factors <= prime(n+1) are {k|k^f(n) mod primorial(n)=1}, where f(n) = lcm(prime(i)-1, i=1..n) = A058254(n) and primorial(n) = A002110(n). For example, numbers with no prime divisor <= prime(7) = 17 are {k|k^60 mod 30030=1}. - Gary Detlefs, Jun 07 2014
Cramer conjecture prime(n+1) - prime(n) < C log^2 prime(n) is equivalent to the inequality (log prime(n+1)/log prime(n))^n < e^C, as n tend to infinity, where C is an absolute constant. - Thomas Ordowski, Oct 06 2014
I conjecture that for any positive rational number r there are finitely many primes q_1,...,q_k such that r = Sum_{j=1..k} 1/(q_j-1). For example, 2 = 1/(2-1) + 1/(3-1) + 1/(5-1) + 1/(7-1) + 1/(13-1) with 2, 3, 5, 7 and 13 all prime, 1/7 = 1/(13-1) + 1/(29-1) + 1/(43-1) with 13, 29 and 43 all prime, and 5/7 = 1/(3-1) + 1/(7-1) + 1/(31-1) + 1/(71-1) with 3, 7, 31 and 71 all prime. - Zhi-Wei Sun, Sep 09 2015
I also conjecture that for any positive rational number r there are finitely many primes p_1,...,p_k such that r = Sum_{j=1..k} 1/(p_j+1). For example, 1 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(11+1) + 1/(23+1) with 2, 3, 5, 7, 11 and 23 all prime, and 10/11 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(43+1) + 1/(131+1) + 1/(263+1) with 2, 3, 5, 7, 43, 131 and 263 all prime. - Zhi-Wei Sun, Sep 13 2015
Numbers k such that ((k-2)!!)^2 == +-1 (mod k). - Thomas Ordowski, Aug 27 2016
Does not satisfy Benford's law [Diaconis, 1977; Cohen-Katz, 1984; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 07 2017
Prime numbers are the integer roots of 1 - sin(Pi*Gamma(s)/s)/sin(Pi/s). - Peter Luschny, Feb 23 2018
Conjecture: log log a(n+1) - log log a(n) < 1/n. - Thomas Ordowski, Feb 17 2023

			From _David A. Corneth_, Oct 22 2024: (Start)
7 is a prime number as it has exactly two divisors, 1 and 7.
8 is not a prime number as it does not have exactly two divisors (it has 1, 2, 4 and 8 as divisors though it is sufficient to find one other divisor than 1 and 8)
55 is not a prime number as it does not have exactly two divisors. One other divisor than 1 and 55 is 5.
59 is a prime number as it has exactly two divisors; 1 and 59. (End)

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.
D. M. Bressoud, Factorization and Primality Testing, Springer-Verlag NY 1989.
M. Cipolla, "La determinazione asintotica dell'n-mo numero primo.", Rend. d. R. Acc. di sc. fis. e mat. di Napoli, s. 3, VIII (1902), pp. 132-166.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 127-149.
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 1.
Martin Davis, "Algorithms, Equations, and Logic", pp. 4-15 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
J.-P. Delahaye, Merveilleux nombres premiers, Pour la Science-Belin Paris, 2000.
J.-P. Delahaye, Savoir si un nombre est premier: facile, Pour La Science, 303(1) 2003, pp. 98-102.
M. Dietzfelbinger, Primality Testing in Polynomial Time, Springer NY 2004.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 5.
J. Elie, "L'algorithme AKS", in 'Quadrature', No. 60, pp. 22-32, 2006 EDP-sciences, Les Ulis (France);
W. & F. Ellison, Prime Numbers, Hermann Paris 1985
T. Estermann, Introduction to Modern Prime Number Theory, Camb. Univ. Press, 1969.
J. M. Gandhi, Formulae for the nth prime. Proc. Washington State Univ. Conf. on Number Theory, 96-106. Wash. St. Univ., Pullman, Wash., 1971.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 77-78.
R. K. Guy, Unsolved Problems Number Theory, Section A.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, pp. (260-264).
H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035, cf. http://www.ams.org/mathscinet-getitem?mr=1336709
M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972.
D. S. Jandu, Prime Numbers And Factorization, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, NY, 1974.
D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Chap. 6.
H. Lifchitz, Table des nombres premiers de 0 à 20 millions (Tomes I & II), Albert Blanchard, Paris 1971.
R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082, cf http://www.ams.org/mathscinet-getitem?mr=96m:11082
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1995.
P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.
H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser Boston, Cambridge MA 1994.
B. Rittaud, "31415879. Ce nombre est-il premier?" ['Is this number prime?'], La Recherche, Vol. 361, pp. 70-73, Feb 15 2003, Paris.
D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, Chap. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 107-119.
D. Wells, Prime Numbers: The Most Mysterious Figures In Math, J. Wiley NY 2005.
H. C. Williams and Jeffrey Shallit, Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143

N. J. A. Sloane, Table of n, prime(n) for n = 1..10000
N. J. A. Sloane, Table of n, prime(n) for n = 1..100000
M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, Annals of Maths., 160:2 (2004), pp. 781-793. [alternative link]
M. Agrawal, A Short History of "PRIMES is in P"
P. Alfeld, Notes and Literature on Prime Numbers
J. W. Andrushkiw, R. I. Andrushkiw and C. E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
Anonymous, Prime Number Master Index (for primes up to 2*10^7)
Anonymous, prime number
Juan Arias de Reyna and Jeremy Toulisse, The n-th prime asymptotically, arxiv:1203.5413 [math.NT], 2012.
Christian Axler, New estimates for the n-th prime number, arXiv:1706.03651 [math.NT], 2017.
P. T. Bateman & H. G. Diamond, A Hundred Years of Prime Numbers, Amer. Math. Month., Vol. 103 (9), Nov. 1996, pp. 729-741.
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64: 2 (2017), 132-134.
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
D. J. Bernstein, Proving Primality After Agrawal-Kayal-Saxena
D. J. Bernstein, Distinguishing prime numbers from composite numbers
P. Berrizbeitia, Sharpening "Primes is in P" for a large family of numbers, arXiv:math/0211334 [math.NT], 2002.
A. Booker, The Nth Prime Page
F. Bornemann, PRIMES Is in P: A Breakthrough for "Everyman", Notices, Amer. Math. Soc., 50: 5 (2003), 545-552.
A. Bowyer, Formulae for Primes
B. M. Bredikhin, Prime number
R. P. Brent, Primality testing and integer factorization
J. Britton, Prime Number List [Dead link]
D. Butler, The first 2000 Prime Numbers
C. K. Caldwell, The Prime Pages: Tables of primes; Lists of small primes (from the first 1000 primes to all 50,000,000 primes up to 982,451,653.)
C. K. Caldwell, A Primality Test
C. K. Caldwell and Y. Xiong, What is the smallest prime?, J. Integer Seq. 15 (2012), no. 9, Article 12.9.7 and arXiv:1209.2007 [math.HO], 2012.
Chris K. Caldwell, Angela Reddick, Yeng Xiong and Wilfrid Keller, The History of the Primality of One: A Selection of Sources, Journal of Integer Sequences, Vol. 15 (2012), #12.9.8.
Ernesto Cesàro, Sur une formule empirique de M. Pervouchine, Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 119 (1894), 848-849.
M. Chamness, Prime number generator (Applet)
Daniel I. A. Cohen and Talbot M. Katz, Prime numbers and the first digit phenomenon, J. Number Theory 18 (1984), 261-268.
P. Cox, Primes is in P
P. J. Davis & R. Hersh, The Mathematical Experience, The Prime Number Theorem
J.-M. De Koninck, Les nombres premiers: mystères et consolation.
J.-M. De Koninck, Nombres premiers: mystères et enjeux.
J.-P. Delahaye, Formules et nombres premiers.
Persi Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probability, 5, 1977, 72--81.
U. Dudley, Formulas for primes, Math. Mag., 56 (1983), 17-22.
Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Thèse, Université de Limoges, France, (1998).
Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68: (1999), 411-415.
J. Elie, L'algorithme AKS ou Les nombres premiers sont de classe P
Seymour B. Elk, Prime Number Assignment to a Hexagonal Tessellation of a Plane That Generates Canonical Names for Peri-Condensed Polybenzenes, J. Chem. Inf. Comput. Sci., vol. 34 (1994), pp. 942-946.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
W. Fendt, Table of Primes from 1 to 1000000000000
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.
J. Flamant, Primes up to one million
K. Ford, Expositions of the PRIMES is in P theorem.
H. Furstenberg, On the Infinitude of Primes, The American Mathematical Monthly, Vol. 62, No. 5 (May, 1955), p. 353 (1 page).
L. & Y. Gallot, The Chronology of Prime Number Records
P. Garrett, Big Primes, Factoring Big Integers
P. Garrett, Naive Primality Test
P. Garrett, Listing Primes
N. Gast, PRIMES is in P: Manindra Agrawal, Neeraj Kayal and Nitin Saxena (in French)
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005.
S. W. Golomb, A Direct Interpretation of Gandhi's Formula, Mathematics Magazine, Vol. 81, No. 7 (Aug. - Sep., 1974), pp. 752-754.
A. Granville, It is easy to determine whether a given integer is prime [alternate link]
G. Hartl Watters, All prime numbers below 2 trillion (compressed `.txt.xz` file). SHA-512 checksum.
P. Hartmann, Prime number proofs (in German) [broken link]
Haskell Wiki, Prime Numbers
ICON Project, List of first 50000 primes grouped within ten columns
James P. Jones, Daihachiro Sato, Hideo Wada and Douglas Wiens, Diophantine representation of the set of prime numbers, The American Mathematical Monthly 83, no. 6 (1976): 449-464. DOI: 10.2307/2318339.
N. Kayal and N. Saxena, A polynomial time algorithm to test if a number is prime or not, Resonance 11-2002.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
W. Liang & H. Yan, Pseudo Random test of prime numbers, arXiv:math/0603450 [math.NT], 2006.
J. Malkevitch, Primes
Mathworld Headline News, Primality Testing is Easy
K. Matthews, Generating prime numbers
Y. Motohashi, Prime numbers-your gems, arXiv:math/0512143 [math.HO], 2005.
Kaoru Motose, On values of cyclotomic polynomials. II, Math. J. Okayama Univ. 37 (1995), 27-36.
J. Moyer, Some Prime Numbers
C. W. Neville, New Results on Primes from an Old Proof of Euler's, arXiv:math/0210282 [math.NT], 2002-2003.
L. C. Noll, Prime numbers, Mersenne Primes, Perfect Numbers, etc.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australasian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
J. J. O'Connor & E. F. Robertson, Prime Numbers
M. E. O'Neill, The Genuine Sieve of Eratosthenes, J. of Functional Programming, Vol 19 Issue 1, Jan 2009, p. 95ff, CUP NY
M. Ogihara & S. Radziszowski, Agrawal-Kayal-Saxena Algorithm for Testing Primality in Polynomial Time
P. Papaphilippou, Plotter of prime numbers frequency graph (flash object) [From Philippos Papaphilippou (philippos(AT)safe-mail.net), Jun 02 2010]
J. M. Parganin, Primes less than 50000
Matthew Parker, The first billion primes (7-Zip compressed file) [a large file]
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
I. Peterson, Prime Pursuits
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Sobre el patrón de los números primos, and from Jason Davies, An interactive companion (for primes 2..997)
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. [Annotated and scanned copy]
Primefan, The First 500 Prime Numbers and Script to Calculate Prime Numbers.
Project Gutenberg Etext, First 100,000 Prime Numbers
C. D. Pruitt, Formulae for Generating All Prime Numbers
R. Ramachandran, Frontline 19 (17) 08-2000, A Prime Solution
W. S. Renwick, EDSAC log.
F. Richman, Generating primes by the sieve of Eratosthenes
Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers, American Journal of Mathematics 63 (1941) 211-232.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. Volume 6, Issue 1 (1962), 64-94.
S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002-2014.
S. O. S. Math, First 1000 Prime Numbers
A. Schulman, Prime Number Calculator
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
M. Slone, PlanetMath.Org, First thousand positive prime numbers
A. Stiglic, The PRIMES is in P little FAQ
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory, 133 (2013), no. 8, 2794-2812.
Zhi-Wei Sun, A conjecture on unit fractions involving primes, Preprint 2015.
J. Teitelbaum, Review of "Prime numbers:A computational perspective" by R. Crandall & C. Pomerance
J. Thonnard, Les nombres premiers(Primality check; Closest next prime; Factorizer)
J. Tramu, Movie of primes scrolling
A. Turpel, Aesthetics of the Prime Sequence [broken link ?]
S. Wagon, Prime Time: Review of "Prime Numbers:A Computational Perspective" by R. Crandall & C. Pomerance
M. R. Watkins, unusual and physical methods for finding prime numbers
S. Wedeniwski, Primality Tests on Commutator Curves
E. Wegrzynowski, Les formules simples qui donnent des nombres premiers en grande quantité (in French).
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial, Prime Number, and Prime Spiral.
Wikipedia, Prime number and Prime number theorem.
C. P. Willans, On formulae for the nth prime, Math. Gazette 48 (1964), 413-415.
G. Xiao, Primes server, Sequential Batches Primes Listing (up to orders not exceeding 10^308)
G. Xiao, Numerical Calculator, To display p(n) for n up to 41561, operate on "prime(n)"
Index entries for "core" sequences
Index entries for sequences related to Benford's law
Additional (less important) items -- comments, formulas, references, links, programs, etc. -- related to the prime numbers, A000040. [HTML version] - [Plain text (TXT) version].

For is_prime and next_prime, see A010051 and A151800.
Cf. A000720 ("pi"), A001223 (differences between primes), A002476, A002808, A003627, A006879, A006880, A008578, A080339, A233588.
Cf. primes in lexicographic order: A210757, A210758, A210759, A210760, A210761.
Cf. A003558, A179480 (relating to the Quasi-order theorem of Hilton and Pedersen).
Boustrophedon transforms: A000747, A000732, A230953.
a(2n) = A104272(n) - A233739(n).
Related sequences:
Primes (p) and composites (c): A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A000040:=Filtered([1..10^5],IsPrime); # Muniru A Asiru, Sep 04 2017

-- See also Haskell Wiki Link.
import Data.List (genericIndex)
a000040 n = genericIndex a000040_list (n - 1)
a000040_list = base ++ larger where
base = [2,3,5,7,11,13,17]
larger = p : filter prime more
prime n = all ((> 0) . mod n) $ takeWhile (\x -> x*x <= n) larger
_ : p : more = roll $ makeWheels base
roll (Wheel n rs) = [n * k + r | k <- [0..], r <- rs]
makeWheels = foldl nextSize (Wheel 1 [1])
nextSize (Wheel size bs) p = Wheel (size * p)
[r | k <- [0..p-1], b <- bs, let r = size*k+b, mod r p > 0]
data Wheel = Wheel Integer [Integer]
-- Reinhard Zumkeller, Apr 07 2014

Magma
```
[n : n in [2..500] | IsPrime(n)];
```
Magma
```
a := func< n | NthPrime(n) >;
```

A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
# For illustration purposes only:
isPrime := s -> is(1 = sin(Pi*GAMMA(s)/s)/sin(Pi/s)):
select(isPrime, [$2..100]); # Peter Luschny, Feb 23 2018

Mathematica
```
Prime[Range[60]]
```

A000040(n) := block(
if n = 1 then return(2),
return( next_prime(A000040(n-1)))
)$ /* recursive, to be replaced if possible - R. J. Mathar, Feb 27 2012 */

PARI
```
{a(n) = if( n<1, 0, prime(n))};
```

/* The following functions provide asymptotic approximations, one based on the asymptotic formula cited above (slight overestimate for n > 10^8), the other one based on pi(x) ~ li(x) = Ei(log(x)) (slight underestimate): */
prime1(n)=n*(log(n)+log(log(n))-1+(log(log(n))-2)/log(n)-((log(log(n))-6)*log(log(n))+11)/log(n)^2/2)
prime2(n)=solve(X=n*log(n)/2,2*n*log(n),real(eint1(-log(X)))+n)
\\ M. F. Hasler, Oct 21 2013

forprime(p=2, 10^3, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

primes(10^5) \\ Altug Alkan, Mar 26 2018

from sympy import primerange
print(list(primerange(2, 272))) # Michael S. Branicky, Apr 30 2022

a = sloane.A000040
a.list(58)  # Jaap Spies, 2007

prime_range(1, 300)  # Zerinvary Lajos, May 27 2009

The prime number theorem is the statement that a(n) ~ n * log n as n -> infinity (Hardy and Wright, page 10).
For n >= 2, n*(log n + log log n - 3/2) < a(n); for n >= 20, a(n) < n*(log n + log log n - 1/2). [Rosser and Schoenfeld]
For all n, a(n) > n log n. [Rosser]
n log(n) + n (log log n - 1) < a(n) < n log n + n log log n for n >= 6. [Dusart, quoted in the Wikipedia article]
a(n) = n log n + n log log n + (n/log n)*(log log n - log n - 2) + O( n (log log n)^2/ (log n)^2). [Cipolla, see also Cesàro or the "Prime number theorem" Wikipedia article for more terms in the expansion]
a(n) = 2 + Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(pi(k)/n)), for n > 1, where the formula for pi(k) is given in A000720 (Ruiz and Sondow 2002). - Jonathan Sondow, Mar 06 2004
I conjecture that Sum_{i>=1} (1/(prime(i)*log(prime(i)))) = Pi/2 = 1.570796327...; Sum_{i=1..100000} (1/(prime(i)*log(prime(i)))) = 1.565585514... It converges very slowly. - Miklos Kristof, Feb 12 2007
The last conjecture has been discussed by the math.research newsgroup recently. The sum, which is greater than Pi/2, is shown in sequence A137245. - T. D. Noe, Jan 13 2009
A000005(a(n)) = 2; A002033(a(n+1)) = 1. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) = 1. - Juri-Stepan Gerasimov, Nov 10 2009
From Gary Detlefs, Sep 10 2010: (Start)
Conjecture:
a(n) = {n| n! mod n^2 = n(n-1)}, n <> 4.
a(n) = {n| n!*h(n) mod n = n-1}, n <> 4, where h(n) = Sum_{k=1..n} 1/k. (End)
For n = 1..15, a(n) = p + abs(p-3/2) + 1/2, where p = m + int((m-3)/2), and m = n + int((n-2)/8) + int((n-4)/8). - Timothy Hopper, Oct 23 2010
a(2n) <= A104272(n) - 2 for n > 1, and a(2n) ~ A104272(n) as n -> infinity. - Jonathan Sondow, Dec 16 2013
Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(n-1) mod n = 1}. - Gary Detlefs, May 25 2014
Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(3*n) mod 3*n = 8}. - Gary Detlefs, May 28 2014
Satisfies a(n) = 2*n + Sum_{k=1..(a(n)-1)} cot(k*Pi/a(n))*sin(2*k*n^a(n)*Pi/a(n)). - Ilya Gutkovskiy, Jun 29 2016
Sum_{n>=1} 1/a(n)^s = P(s), where P(s) is the prime zeta function. - Eric W. Weisstein, Nov 08 2016
a(n) = floor(1 - log(-1/2 + Sum_{ d | A002110(n-1) } mu(d)/(2^d-1))/log(2)) where mu(d) = A008683(d) [Ghandi, 1971] (see Ribenboim). Golomb gave a proof in 1974: Give each positive integer a probability of W(n) = 1/2^n, then the probability M(d) of the integer multiple of number d equals 1/(2^d-1). Suppose Q = a(1)*a(2)*...*a(n-1) = A002110(n-1), then the probability of random integers that are mutually prime with Q is Sum_{ d | Q } mu(d)*M(d) = Sum_{ d | Q } mu(d)/(2^d-1) = Sum_{ gcd(m, Q) = 1 } W(m) = 1/2 + 1/2^a(n) + 1/2^a(n+1) + 1/2^a(n+2) + ... So ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) = 1 + x(n), which means that a(n) is the only integer so that 1 < ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) < 2. - Jinyuan Wang, Apr 08 2019
Conjecture: n * (log(n)+log(log(n))-1+((log(log(n))-A)/log(n))) is asymptotic to a(n) if and only if A=2. - Alain Rocchelli, Feb 12 2025
From Stefano Spezia, Apr 13 2025: (Start)
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/Sum_{j=1..m} A080339(j))^(1/n)) [Willans, 1964].
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/(1 + A000720(m)))^(1/n)) [Willans, 1964]. (End)

A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 1

N. J. A. Sloane

Partial sums of A010051 (characteristic function of primes). - Jeremy Gardiner, Aug 13 2002
pi(n) and prime(n) are inverse functions: a(A000040(n)) = n and A000040(n) is the least number m such that A000040(a(m)) = A000040(n). A000040(a(n)) = n if (and only if) n is prime. - Jonathan Sondow, Dec 27 2004
See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
A lower bound that gets better with larger N is that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i=0..T(sqrt(N))} A005867(i)/A002110(i). - Ben Paul Thurston, Aug 23 2010
Number of partitions of 2n into exactly two parts with the smallest part prime. - Wesley Ivan Hurt, Jul 20 2013
Equivalent to the Riemann hypothesis: abs(a(n) - li(n)) < sqrt(n)*log(n)/(8*Pi), for n >= 2657, where li(n) is the logarithmic integral (Lowell Schoenfeld). - Ilya Gutkovskiy, Jul 05 2016
The second Hardy-Littlewood conjecture, that pi(x) + pi(y) >= pi(x + y) for integers x and y with min{x, y} >= 2, is known to hold for (x, y) sufficiently large (Udrescu 1975). - Peter Luschny, Jan 12 2021

			There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 8.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 409.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 5.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorems 6, 7, 420.
G. J. O. Jameson, The Prime Number Theorem, Camb. Univ. Press, 2003. [See also the review by D. M. Bressoud (link below).]
Władysław Narkiewicz, The Development of Prime Number Theory, Springer-Verlag, 2000.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 132-133, 157-184.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.1. (For inequalities, etc.).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gerald Tenenbaum and Michel Mendès France, Prime Numbers and Their Distribution, AMS Providence RI, 1999.
V. Udrescu, Some remarks concerning the conjecture pi(x + y) <= pi(x) + pi(y), Rev. Roumaine Math. Pures Appl. 20 (1975), 1201-1208.

Daniel Forgues, Table of n, pi(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane; see below for links with 823852 terms (Verma) and more (Caldwell))
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Christian Axler, Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen, Ph.D. thesis 2013, in German, English summary.
Paul T. Bateman and Harold G. Diamond, A Hundred Years of Prime Numbers, Amer. Math. Month., Vol. 103, No. 9 (Nov. 1996), pp. 729-741, MAA Washington DC.
Steve Bennett, The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem, 2004.
Claudio Bonanno and Mirko S. Mega, Toward a dynamical model for prime numbers, Chaos, Solitons & Fractals, Vol. 20, No. 1 (2004), pp. 107-118; arXiv preprint, arXiv:cond-mat/0309251, 2003.
David M. Bressoud, Review of "The Prime Number Theorem" by G. J. O. Jameson, MAA Reviews, 2005. - from _N. J. A. Sloane_, Dec 29 2018
D. M. Bressoud and Stan Wagon, Computational Number Theory: Basic Algorithms, Springer/Key, 2000 (with a Mathematica package for computational number theory).
Ben Brubaker, The Prime Number Theorem.
C. K. Caldwell, The Prime Glossary: Prime number theorem.
W. W. L. Chen, Distribution of Prime Numbers.
Jean-Marie de Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See Chapter 9, p. 231.
Marc Deléglise, Computation of large values of pi(x), 1996.
Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Thèse, Université de Limoges, France (1998).
Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation, Vo. 68, No. 225 (1999), pp. 411-415.
Encyclopedia Britannica, The Prime Number Theorem [web.archive.org's copy of a no longer available personal copy of the Encyclopedia's article]
R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Mathematica, Vol. 41 (1916), pp. 119-196.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Mehdi Hassani, Approximation of pi(x) by Psi(x), J. Inequ. Pure Appl. Math., Vol. 7, No. 1 (2006), Article #7.
Y.-C. Kim, Note on the Prime Number Theorem, arXiv:math/0502062 [math.NT], 2005.
Tzanio V. Kolev, On the number of Prime Numbers less than a Given Quantity, 2000.
Angel V. Kumchev, The Distribution of Prime Numbers, 2005.
J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., Vol. 44, No. 170 (1985), pp. 537-560.
Jeffrey C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, Journal of Algorithms, Vol. 8, No. 2 (1987), pp. 173-191; alternative link.
John Lorch, The Distribution of Primes, B.S. Undergraduate Mathematics Exchange, Vol. 3, No. 1 (Fall 2005).
Nathan McKenzie, Computing the Prime Counting Function with Linnik's Identity, personal blog, March 24, 2011.
Murat Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.
Bent E. Petersen, Prime Number Theorem, Seminar Lecture Note, 1996; version 2002-05-14.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos, personal website, 2001 (?); Illustration of initial terms: Divisors and pi(x).
Bernhard Riemann, On the Number of Prime Numbers, 1859, last page (various transcripts)
J. Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers, American Journal of Mathematics, Vol. 63, No. 1 (1941), pp. 211-232.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy).
Sebastian Martin Ruiz and Jonathan Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002, 2014.
Jeffrey Shallit, Bibliography on calculation of pi(x).
Jonathan Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly, Vol. 116, No. 7 (2009), pp. 630-635; arXiv preprint, arXiv:0907.5232 [math.NT], 2009-2010.
Igor Turkanov, The prime counting function, arXiv:1603.02914 [math.NT], 2016.
Gaurav Verma and Srujan Sapkal, Table of n, pi(n) for n = 1..823852.
Matthew R. Watkins, The distribution of Prime Numbers.
Matthew R. Watkins, The prime number theorem (some references).
Eric Weisstein's World of Mathematics, Prime Counting Function.
Wikipedia, Prime number theorem.
Wikipedia, Prime-counting function.
C. P. Willans, On formulae for the nth prime, Math. Gazette 48 (1964), 413-415.
Marek Wolf, Applications of Statistical Mechanics in Number Theory, Physica A, vol. 274, no. 2, 1999, pp. 149-157; 1999 preprint.
Wolfram Research, First 50 values of pi(n).
Index entries for "core" sequences

Cf. A048989, A000040, A132090, A137588, A139328, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A143538, A036234, A033844, A034387, A034386, A179215, A010051, A212210-A212213, A233824, A056171, A304483.
Closely related:
A099802: Number of primes <= 2n.
A060715: Number of primes between n and 2n (exclusive).
A035250: Number of primes between n and 2n (inclusive).
A038107: Number of primes < n^2.
A014085: Number of primes between n^2 and (n+1)^2.
A007053: Number of primes <= 2^n.
A036378: Number of primes p between powers of 2, 2^n < p <= 2^(n+1).
A006880: Number of primes < 10^n.
A006879: Number of primes with n digits.
A033270: Number of odd primes <= n.
A065855: Number of composites <= n.
For lists of large values of a(n) see, e.g., A005669(n) = a(A002386(n)), A214935(n) = a(A205827(n)).
Related sequences:
Primes (p) and composites (c): A000040, A002808, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a000720 n = a000720_list !! (n-1)
a000720_list = scanl1 (+) a010051_list  -- Reinhard Zumkeller, Sep 15 2011

[ #PrimesUpTo(n): n in [1..200] ];  // Bruno Berselli, Jul 06 2011

with(numtheory); A000720 := pi; [ seq(A000720(i),i=1..50) ];

A000720[n_] := PrimePi[n]; Table[ A000720[n], {n, 1, 100} ]
Array[ PrimePi[ # ]&, 100 ]
Accumulate[Table[Boole[PrimeQ[n]],{n,100}]] (* Harvey P. Dale, Jan 17 2015 *)

A000720=vector(100,n,omega(n!)) \\ For illustration only; better use A000720=primepi

vector(300,j,primepi(j)) \\ Joerg Arndt, May 09 2008

from sympy import primepi
for n in range(1,100): print(primepi(n), end=', ') # Stefano Spezia, Nov 30 2018

[prime_pi(n) for n in range(1, 79)]  # Zerinvary Lajos, Jun 06 2009

The prime number theorem gives the asymptotic expression a(n) ~ n/log(n).
For x > 1, pi(x) < (x / log x) * (1 + 3/(2 log x)). For x >= 59, pi(x) > (x / log x) * (1 + 1/(2 log x)). [Rosser and Schoenfeld]
For x >= 355991, pi(x) < (x / log(x)) * (1 + 1/log(x) + 2.51/(log(x))^2 ). For x >= 599, pi(x) > (x / log(x)) * (1 + 1/log(x)). [Dusart]
For x >= 55, x/(log(x) + 2) < pi(x) < x/(log(x) - 4). [Rosser]
For n > 1, A138194(n) <= a(n) <= A138195(n) (Tschebyscheff, 1850). - Reinhard Zumkeller, Mar 04 2008
For n >= 33, a(n) = 1 + Sum_{j=3..n} ((j-2)! - j*floor((j-2)!/j)) (Hardy and Wright); for n >= 1, a(n) = n - 1 + Sum_{j=2..n} (floor((2 - Sum_{i=1..j} (floor(j/i)-floor((j-1)/i)))/j)) (Ruiz and Sondow 2000). - Benoit Cloitre, Aug 31 2003
a(n) = A001221(A000142(n)). - Benoit Cloitre, Jun 03 2005
G.f.: Sum_{p prime} x^p/(1-x) = b(x)/(1-x), where b(x) is the g.f. for A010051. - Franklin T. Adams-Watters, Jun 15 2006
a(n) = A036234(n) - 1. - Jaroslav Krizek, Mar 23 2009
From Enrique Pérez Herrero, Jul 12 2010: (Start)
a(n) = Sum_{i=2..n} floor((i+1)/A000203(i)).
a(n) = Sum_{i=2..n} floor(A000010(n)/(i-1)).
a(n) = Sum_{i=2..n} floor(2/A000005(n)). (End)
Let pf(n) denote the set of prime factors of an integer n. Then a(n) = card(pf(n!/floor(n/2)!)). - Peter Luschny, Mar 13 2011
a(n) = -Sum_{p <= n} mu(p). - Wesley Ivan Hurt, Jan 04 2013
a(n) = (1/2)*Sum_{p <= n} (mu(p)*d(p)*sigma(p)*phi(p)) + sum_{p <= n} p^2. - Wesley Ivan Hurt, Jan 04 2013
a(1) = 0 and then, for all k >= 1, repeat k A001223(k) times. - Jean-Christophe Hervé, Oct 29 2013
a(n) = n/(log(n) - 1 - Sum_{k=1..m} A233824(k)/log(n)^k + O(1/log(n)^{m+1})) for m > 0. - Jonathan Sondow, Dec 19 2013
a(n) = A001221(A003418(n)). - Eric Desbiaux, May 01 2014
a(n) = Sum_{j=2..n} H(-sin^2 (Pi*(Gamma(j)+1)/j)) where H(x) is the Heaviside step function, taking H(0)=1. - Keshav Raghavan, Jun 18 2016
a(A014076(n)) = (1/2) * (A014076(n) + 1) - n + 1. - Christopher Heiling, Mar 03 2017
From Steven Foster Clark, Sep 25 2018: (Start)
a(n) = Sum_{m=1..n} A143519(m) * floor(n/m).
a(n) = Sum_{m=1..n} A001221(m) * A002321(floor(n/m)) where A002321() is the Mertens function.
a(n) = Sum_{m=1..n} |A143519(m)| * A002819(floor(n/m)) where A002819() is the Liouville Lambda summatory function and |x| is the absolute value of x.
a(n) = Sum_{m=1..n} A137851(m)/m * H(floor(n/m)) where H(n) = Sum_{m=1..n} 1/m is the harmonic number function.
a(n) = Sum_{m=1..log_2(n)} A008683(m) * A025528(floor(n^(1/m))) where A008683() is the Moebius mu function and A025528() is the prime-power counting function.
(End)
Sum_{k=2..n} 1/a(k) ~ (1/2) * log(n)^2 + O(log(n)) (de Koninck and Ivić, 1980). - Amiram Eldar, Mar 08 2021
a(n) ~ 1/(n^(1/n)-1). - Thomas Ordowski, Jan 30 2023
a(n) = Sum_{j=2..n} floor(((j - 1)! + 1)/j - floor((j - 1)!/j)) [Mináč, unpublished] (see Ribenboim, pp. 132-133). - Stefano Spezia, Apr 13 2025
a(n) = n - 1 - Sum_{k=2..floor(log_2(n))} pi_k(n), where pi_k(n) is the number of k-almost primes <= n. - Daniel Suteu, Aug 27 2025

Additional links contributed by Lekraj Beedassy, Dec 23 2003

Edited by M. F. Hasler, Apr 27 2018 and (links recovered) Dec 21 2018

A006880 Number of primes < 10^n.

Original entry on oeis.org

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

Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to (10^n)/log(10^n) (see A228065).
The sequence A006879(n) is always > A228065(n) for 1 <= n <= 28.
The sequence (A228065) provides exactly the first value of pi(10^n)- pi(10^(n-1)) for n = 1, and yields an average relative difference in absolute value, i.e., average(abs(A228066(n))/(A006879(n))) = 0.0436296... for 1 <= n <= 28.
Note that A057834(n) = 10^n/log(10^n) is not defined for n = 0; its value is set arbitrarily to 0. - Updated by Eduard Roure Perdices, Apr 18 2021

Difference between the number of primes with n digits (A006879) and its estimate by squares of odd-indexed Fibonacci polynomials (A228115).
The sequence (A228115) provides exactly the values of pi(10^n)- pi(10^(n-1)) for n=2 to 4 and yields an average relative difference in absolute value, i.e. Average(Abs(A228116(n))/ (A006879(n)) = 1.01656…x10^-2 for 1<=n<=25, better than when using the ((10^n)/log(10^n)) function (Average(Abs(A228066(n))/ (A006879(n)) = 4.69094…x10^-2 (see A228066)), or the Logarithm integral (Li(10^n)-Li(2)) function (Average(Abs(A228068(n))/ (A006879(n)) = 1.75492…x10^-2 (see A228068)), or the Riemann(Riemann (10^n)) function (Average(Abs(A228114(n))/ (A006879(n)) = 1.03936…x10^-2) for 1<=n<=25.
Furthermore, if the first value for n=1 is skipped, the average relative difference in absolute value is improved by nearly two orders of magnitude, i.e. Average(Abs(A228116(n))/ (A006879(n)) = 1.72564…x10^-4 for 2<=n<=25, better than when using the ((10^n)/log(10^n)) function (Average(Abs(A228066(n))/ (A006879(n)) = 4.88640…x10^-2 (see A228066)), or the Logarithm integral (Li(10^n)-Li(2)) function (Average(Abs(A228068(n))/ (A006879(n))  = 7.86383…x10^-3 (see A228068)), or the Riemann(Riemann (10^n)) function (Average(Abs(A228114(n))/ (A006879(n)) = 4.10042…x10^-4), or the product of x and Fibonacci polynomials of multiple of 4 indices F[4n](x) (Average(Abs(A228064(n))/ (A006879(n)) = 3.90981…x10^-3 (see A228112)) for 2<=n<=25.

The sequence A006879(n) is always < A228067(n) for 1 <= n <= 25.
The sequence (A228067) yields an average relative difference in absolute value, i.e., average(abs(A228068(n))/A006879(n) = 0.0175492... for 1 <= n <= 25.
Note that A190802(n) = (Li(10^n) - Li(2)) is not defined for n=0. Its value is set arbitrarily to 0.

A228111 provides exact values of pi(10^n) - pi(10^(n-1)) for n = 1 to 3 and yields an average relative difference in absolute value, i.e. average(abs(A228112(n))/A006879(n) = 0.00375341... for 1 <= n <= 25, better than when using the 10^n/log(10^n) function, which yields 0.0469094... (see A228066) or the logarithmic integral (Li(10^n) - Li(2)) function, which yields 0.0175492... (see A228068) or the Riemann (Riemann(10^n)) function, which yields 0.0103936... (see A228114) or the Fibonacci polynomials of multiple of 4 indices, which yields 0.00473860... (see A228064) for 1 <= n <= 25.

The sequence (A228113) yields an average relative difference in absolute value, i.e. Average(Abs(A228114(n))/ (A006879(n)) = 1.03936…x10^-2 for 1<=n<=25.

Note that A057793(n) = Riemann(10^n) is not defined for n=0. Its value is set to 0.

A228063 provides exactly the values of pi(10^n) - pi(10^(n-1)) for n = 1 to 3 and yields an average relative difference in absolute value, i.e., average(abs(A228064(n))/A006879(n) = 0.00473860... for 1 <= n <= 25, better than when using the 10^n/log(10^n) function, which yields 0.0469094... (see A228066) or the logarithmic integral (Li(10^n) - Li(2)) function, which yields 0.0175492... (see A228068) for 1 <= n <= 25.

cp's OEIS Frontend

A228066 a(n) = A006879(n) - A228065(n).

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A228116 a(n) = A006879(n) - A228115(n).

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A228068 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Li(10^n) - Li(2) (see A228067).

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A228112 Difference between the number of primes with n digits (A006879) and the 6-parametric approximation of that number in A228111.

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A228114 Difference between the number of primes with n digits (A006879) and the difference of consecutive integers nearest to Riemann(10^n) (see A228113).

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A228064 Difference between the number of primes with n digits (A006879) and the nearest integer to F[4n](S(n)), where F[4n](x) are Fibonacci polynomials and S(n) = Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i) (see A228063).

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A000040 The prime numbers.

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A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...

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A228064 Difference between the number of primes with n digits (A006879) and the nearest integer to F[4n](S(n)), where F[4n](x) are Fibonacci polynomials and S(n) = Sum_{i=0..3} (C(i)(log(log(A(B+n^2))))^i) (see A228063).