A006880 -id:A006880 - 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)

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M. Agrawal, A Short History of "PRIMES is in P"
P. Alfeld, Notes and Literature on Prime Numbers
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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.
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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

A007053 Number of primes <= 2^n.

Original entry on oeis.org

0, 1, 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, 155611, 295947, 564163, 1077871, 2063689, 3957809, 7603553, 14630843, 28192750, 54400028, 105097565, 203280221, 393615806, 762939111, 1480206279, 2874398515, 5586502348, 10866266172, 21151907950, 41203088796, 80316571436, 156661034233, 305761713237, 597116381732, 1166746786182, 2280998753949, 4461632979717, 8731188863470, 17094432576778, 33483379603407, 65612899915304, 128625503610475

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, S. W. Golomb

Conjecture: The number 4 is the only perfect power in this sequence. In other words, it is impossible to have a(n) = x^m for some integers n > 3, m > 1 and x > 1. - Zhi-Wei Sun, Sep 30 2015

			pi(2^3)=4 since first 4 primes are 2,3,5,7 all <= 2^3 = 8.

Jens Franke et al., pi(10^24), Posting to the Number Theory Mailing List, Jul 29 2010.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Baugh, Table of n, a(n) for n = 0..92 (terms n = 87..92 found using Kim Walisch's primecount program, terms n = 0..86 from Charles R Greathouse IV and Douglas B. Staple, [a(0)-a(75) from Tomás Oliveira e Silva, a(76)-a(77) from Jens Franke et al., Jul 29 2010, a(78)-a(80) from Jens Franke et al. on the Riemann Hypothesis, verified unconditionally by Douglas B. Staple, and a(81)-a(86) from Douglas B. Staple])
Andrew R. Booker, The Nth Prime Page
S. W. Golomb, Letter to N. J. A. Sloane, Jul. 1991
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [Local copy, pdf only]
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, No 6, March 2006.
Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015.
Index entries for sequences related to numbers of primes in various ranges

Cf. A006880, A036378.

Table[PrimePi[2^n], {n, 0, 46}] (* Robert G. Wilson v *)

a(n) = primepi(1<John W. Nicholson, May 16 2011

a(n) = A060967(2n). - R. J. Mathar, Sep 15 2012

More terms from Jud McCranie
Extended to n = 52 by Warren D. Smith, Dec 11 2000, computed with Meissel-Lehmer-Legendre inclusion exclusion formula code he wrote back in 1985, recently re-run.
Extended to n = 86 by Douglas B. Staple, Dec 18 2014

A007508 Number of twin prime pairs below 10^n.

Original entry on oeis.org

2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

"At the present time (2001), Thomas Nicely has reached pi_2(3*10^15) and his value is confirmed by Pascal Sebah who made a new computation from scratch and up to pi_2(5*10^15) [ = 5357875276068] with an independent implementation."
Though the first paper contributed by D. A. Goldston was reported to be flawed, the more recent one (with other coauthors) maintains and substantiates the result. - Lekraj Beedassy, Aug 19 2005
Theorem. While g is even, g > 0, number of primes p < x (x is an integer) such that p' = p + g is also prime, could be written as qpg(x) = qcc(x) - (x - pi(x) - pi(x + g) + 1) where qcc(x) is the number of "common composite numbers" c <= x such that c and c' = c + g both are composite (see Example below; I propose it here as a theorem only not to repeat for so-called "cousin"-primes (p; p+4), "sexy"-primes (p; p+6), etc.). - Sergey Pavlov, Apr 08 2021

			For x = 10, qcc(x) = 4 (since 2 is prime; 4, 6, 8, 10 are even, and no odd 0 < d < 25 such that both d and d' = d + 2 are composite), pi(10) = 4, pi(10 + 2) = 5, but, while v = 2+2 or v = 2-2 would be even, we must add 1; hence, a(1) = qcc(10^1) - (10^1 - pi(10^1) - pi(10^1 + 2) + 1) = 4 - (10 - 4 - 5 + 1) = 2 (trivial). - _Sergey Pavlov_, Apr 08 2021

Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 202.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 195.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43-56.
C. K. Caldwell, An amazing prime heuristic, Table 1.
C. K. Caldwell, The Prime Glossary, Twin prime conjecture
T. H. Chan, A note on Primes in Short Intervals, arXiv:math/0503441 [math.NT], 2005.
P. Erdős, Some Unsolved Problems, Michigan Math. J., Volume 4, Issue 3 (1957), 291-300.
G. H. Gadiyar & R. Padma, Renormalisation and the density of prime pairs, arXiv:hep-th/9806061, 1998.
G. H. Gadiyar & R. Padma, Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs, Physica A 269 (1999) 503-510.
D. A. Goldston, J. Pintz & C. Y. Yildirim, Primes in Tuples, I, arXiv:math/0508185 [math.NT], 2005.
D. A. Goldston, J. Pintz & C. Y. Yildirim, Small Gaps Between Primes, II
D. A. Goldston, J. Pintz & C. Y. Yildirim, The Path to Recent Progress on Small Gaps Between Primes, arXiv:math/0512436 [math.NT], 2005-2006.
D. A. Goldston & C. Y. Yildirim, Small Gaps Between Primes, I, arXiv:math/0504336 [math.NT], 2005.
D. A. Goldston & C. Yildirim, Small gaps between consecutive primes
D. A. Goldston et al., Small gaps between primes or almost primes, arXiv:math/0506067 [math.NT], 2005.
D. A. Goldston et al., Small Gaps between Primes Exist, arXiv:math/0505300 [math.NT], 2005.
Xavier Gourdon and Pascal Sebah, Introduction to Twin Primes and Brun's Constant
A. Granville & K. Soundararajan, On the error in Goldston and Yildirim's "Small gaps between consecutive primes"
Gareth A. Jones and Alexander K. Zvonkin, Klein's ten planar dessins of degree 11, and beyond, arXiv:2104.12015 [math.GR], 2021. See p. 24.
P. F. Kelly & F. Pilling, Characterization of the Distribution of Twin Primes, arXiv:math/0103191 [math.NT], 2001.
P. F. Kelly & T. Pilling, Implications of a New Characterization of the Distribution of Twin Primes, arXiv:math/0104205 [math.NT], 2001.
P. F. Kelly & T. Pilling, Discrete Reanalysis of a New Model of the Distribution of Twin Primes, arXiv:math/0106223 [math.NT], 2001.
James Maynard, On the Twin Prime Conjecture, arXiv:1910.14674 [math.NT], 2019, p. 2.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only]
Nova Science, Twin Prime Conjecture
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From _M. F. Hasler_, Dec 18 2008]
J. Richstein, Computing the number of twin primes up to 10^14
J. Richstein, Computing the number of twin primes up to 10^14
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.
K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], 2006.
K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim, arXiv:math/0605696 [math.NT], 2006.
K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), 1-18.
Eric Weisstein's World of Mathematics, Twin Primes
Eric Weisstein, Mathworld Headline News, Twin Primes Proof Proffered
M. Wolf, Some Remarks on the Distribution of twin Primes, arXiv:math/0105211 [math.NT], 2001.
C. Yildirim & D. Goldston, Small gaps between consecutive primes
Index entries for sequences related to numbers of primes in various ranges

Cf. A001097.
Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).
Cf. A152051, A347278, A347279.

ile = 2; Do[Do[If[(PrimeQ[2 n - 1]) && (PrimeQ[2 n + 1]), ile = ile + 1], {n, 5*10^m, 5*10^(m + 1)}]; Print[{m, ile}], {m, 0, 7}] (* Artur Jasinski, Oct 24 2011 *)

a(n)=my(s,p=2);forprime(q=3,10^n,if(q-p==2,s++);p=q);s \\ Charles R Greathouse IV, Mar 21 2013

Partial sums of A070076(n). - Lekraj Beedassy, Jun 11 2004
For 1 < n < 19, a(n) ~ e * pi(10^n) / (5*n - 5) = e * A006880(n) / (5*n - 5) where e is Napier's constant, see A001113 (probably, so is for any n > 18; we use n > 1 to avoid division by zero). - Sergey Pavlov, Apr 07 2021
For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(10^n + 2) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that c and c' = c + 2 both are composite (trivial). - Sergey Pavlov, Apr 08 2021

pi2(10^15) due to Nicely and Szymanski, contributed by Eric W. Weisstein
pi2(10^16) due to Pascal Sebah, contributed by Robert G. Wilson v, Aug 22 2002
Added a(17)-a(18) computed by Tomás Oliveira e Silva and link to his web site. - M. F. Hasler, Dec 18 2008
Definition corrected by Max Alekseyev, Oct 25 2010
a(16) corrected by Dana Jacobsen, Mar 28 2014

As Li(3)= 2.163588..., A057752(n)-a(n) = 2, except for n =3, 6, 10, 11, 15, 20 where A057752(n)-a(n)= 3.

This sequence yields an even better average relative difference than Gauss's approximation (A106313), i.e., Average(a(n)/pi(10^n)) = 7.4969...*10^-3 for 1<=n<=24, compared to Average(A057752(n)/pi(10^n)) = 3.2486...*10^-2 and Average(A106313(n)/pi(10^n)) = 2.0116...*10^-2, showing that, when using the logarithmic integral, Li(10^n)-Li(3) (A223166) gives a better approximation to pi(10^n) than Li(10^n)-Li(2) (A190802) and than Li(10^n) (A057754).

A225137 provides exactly the values of pi(10^n) for n = 1, 2, 3, 5 and 9 and yields an average relative difference in absolute value, i.e., average(abs(A225138(n))/pi(10^n)) = 7.2165...*10^-5 for 1 <= n <= 24.

A225137 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n), whether as the sequence of integers <= R(10^n) (A215663), which yields 1.453...*10^-4, or as the sequence of integers nearest to R(10^n) (A057794), which yields 0.01219...; (2) the functions of the logarithmic integral Li(x) = Integral_{t=0..x} dt/log(t), whether as the sequence of integers nearest to (Li(10^n) - Li(3)) (A223166), which yields 7.4969...x10^-3 (see A223167), or as Gauss's approximation to pi(10^n), i.e., the sequence of integers nearest to (Li(10^n) - Li(2)) (A190802) = 0.020116... (see A106313), or as the sequence of integers nearest to Li(10^n) (A057752), which yields 0.032486....

A227693 provides exactly the values of pi(10^n) for n = 1 to 4 and yields an average relative difference in absolute value, average(abs(A227694(n))/pi(10^n)) = 1.58269...*10^-4 for 1 <= n <= 25.

A227693 provides a better approximation to the distribution of pi(10^n) than: (1) the Riemann function R(10^n) as the sequence of integers nearest to R(10^n) (A057794), which yields 0.01219...; (2) the functions of the logarithmic integral Li(x) = Integral_{t=0..x} dt/log(t), whether as the sequence of integers nearest to (Li(10^n) - Li(3)) (A223166), which yields 0.0074969... (see A223167), or as Gauss's approximation to pi(10^n), i.e., the sequence of integers nearest to (Li(10^n) - Li(2)) (A190802), which yields 0.020116... (see A106313), or as the sequence of integer nearest to Li(10^n) (A057752), which yields 0.032486....

cp's OEIS Frontend

A057752 Difference between nearest integer to Li(10^n) and pi(10^n), where Li(x) = integral of log(x) and pi(10^n) = number of primes <= 10^n (A006880).

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A223167 Difference between nearest integer to (Li(10^n)-Li(3)) and pi(10^n), where Li(10^n)-Li(3) = integral(3.. 10^n, dt/log(t)) (A223166) and pi(10^n) = number of primes <= 10^n (A006880).

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A221205 a(n) is the nearest integer to sqrt(pi(10^n)) (see A006880).

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A225138 Difference between pi(10^n) and nearest integer to (4*((S(n))^(n-1))) where pi(10^n) = number of primes <= 10^n (A006880) and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^(8/3)))))^(2i)) (A225137).

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A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)*(log(log(A*(B+n^2))))^(2i)) (see A227693).

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A135377 Smallest n-primeval prime, i.e., minimal prime number containing all A006880(n) primes < 10^n embedded in it as permutations of some of its substrings.

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A000040 The prime numbers.

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A225138 Difference between pi(10^n) and nearest integer to (4((S(n))^(n-1))) where pi(10^n) = number of primes <= 10^n (A006880) and S(n) = Sum_{i=0..2} (C(i)(log(log(A*(B+n^(8/3)))))^(2i)) (A225137).

A227694 Difference between pi(10^n) and nearest integer to (F[2n+1](S(n)))^2 where pi(10^n) = number of primes <= 10^n (A006880), F[2n+1](x) are Fibonacci polynomials of odd indices [2n+1] and S(n) = Sum_{i=0..2} (C(i)(log(log(A(B+n^2))))^(2i)) (see A227693).