A060715 -id:A060715 - 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
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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

A002808 The composite numbers: numbers n of the form x*y for x > 1 and y > 1.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

The natural numbers 1,2,... are divided into three sets: 1 (the unit), the primes (A000040) and the composite numbers (A002808).
The number of composite numbers <= n (A065855) = n - pi(n) (A000720) - 1.
n is composite iff sigma(n) + phi(n) > 2n. This is a nice result of the well known theorem: For all positive integers n, n = Sum_{d|n} phi(d). For the proof see my contribution to puzzle 76 of Carlos Rivera's Primepuzzles. - Farideh Firoozbakht, Jan 27 2005, Jan 18 2015
The composite numbers have the semiprimes A001358 as primitive elements.
A211110(a(n)) > 1. - Reinhard Zumkeller, Apr 02 2012
A060448(a(n)) > 1. - Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) > 0. - Reinhard Zumkeller, Dec 14 2012
Composite numbers n which are the product of r=A001222(n) prime numbers are sometimes called r-almost primes. Sequences listing r-almost primes are: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
a(n) = A056608(n) * A160180(n). - Reinhard Zumkeller, Mar 29 2014
Degrees for which there are irreducible polynomials which are reducible mod p for all primes p, see Brandl. - Charles R Greathouse IV, Sep 04 2014
An integer is composite if and only if it is the sum of strictly positive integers in arithmetic progression with common difference 2: 4 = 1 + 3, 6 = 2 + 4, 8 = 3 + 5, 9 = 1 + 3 + 5, etc. - Jean-Christophe Hervé, Oct 02 2014
This statement holds since k+(k+2)+...+k+2(n-1) = n*(n+k-1) = a*b with arbitrary a,b (taking n=a and k=b-a+1 if b>=a). - M. F. Hasler, Oct 04 2014
For n > 4, these are numbers n such that n!/n^2 = (n-1)!/n is an integer (see A056653). - Derek Orr, Apr 16 2015
Let f(x) = Sum_{i=1..x} Sum_{j=2..i-1} cos((2*Pi*x*j)/i). It is known that the zeros of f(x) are the prime numbers. So these are the numbers n such that f(n) > 0. - Michel Lagneau, Oct 13 2015
Numbers n that can be written as solutions of the Diophantine equation n = (x+2)(y+2) where {x,y} in N^2, pairs of natural numbers including zero (cf. Mathematica code and Davis). - Ron R Spencer and Bradley Klee, Aug 15 2016
Numbers n with a partition (containing at least two summands) so that its summands also multiply to n. If n is prime, there is no way to find those two (or more) summands. If n is composite, simply take a factor or several, write those divisors and fill with enough 1's so that they add up to n. For example: 4 = 2*2 = 2+2, 6 = 1*2*3 = 1+2+3, 8 = 1*1*2*4 = 1+1+2+4, 9 = 1*1*1*3*3 = 1+1+1+3+3. - Juhani Heino, Aug 02 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 127.
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.
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.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 66.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 51.
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..17737 [composites up to 20000]
Rolf Brandl, Integer polynomials that are reducible modulo all primes, Amer. Math. Monthly 93 (1986), pp. 286-288.
C. K. Caldwell, Composite Numbers
Felix Huber, Illustration for a(1)-a(12)
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
Carlos Rivera, Puzzle 76, z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94
Eric Weisstein's World of Mathematics, Composite Number
Index entries for "core" sequences
Index entries for sequences from "Goedel, Escher, Bach"

Complement of A008578. - Omar E. Pol, Dec 16 2016
Cf. A000040, A018252, A065090, A136527.
Cf. A073783 (first differences), A073445 (second differences).
Boustrophedon transforms: A230954, A230955.
Cf. A163870 (nontrivial divisors).
Related sequences:
Primes (p) and composites (c): A000040, 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.

a002808 n = a002808_list !! (n-1)
a002808_list = filter ((== 1) . a066247) [2..]
-- Reinhard Zumkeller, Feb 04 2012

[n: n in [2..250] | not IsPrime(n)]; // G. C. Greubel, Feb 24 2024

t := []: for n from 2 to 20000 do if isprime(n) then else t := [op(t),n]; fi; od: t; remove(isprime,[$3..89]); # Zerinvary Lajos, Mar 19 2007
A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do ; end if; end proc; # R. J. Mathar, Oct 27 2009

Select[Range[2,100], !PrimeQ[#]&] (* Zak Seidov, Mar 05 2011 *)
With[{nn=100},Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, May 01 2012 *)
Select[Range[100], CompositeQ] (* Jean-François Alcover, Nov 07 2021 *)

A002808(n)=for(k=0,primepi(n),isprime(n++)&&k--);n \\ For illustration only: see below. - M. F. Hasler, Oct 31 2008

A002808(n)= my(k=-1); while(-n + n += -k + k=primepi(n),); n \\ For n=10^4 resp. 3*10^4, this is about 100 resp. 500 times faster than the former; M. F. Hasler, Nov 11 2009

forcomposite(n=1, 1e2, print1(n, ", ")) \\ Felix Fröhlich, Aug 03 2014

for(n=1, 1e3, if(bigomega(n) > 1, print1(n, ", "))) \\ Altug Alkan, Oct 14 2015

from sympy import primepi
def A002808(n):
    m, k = n, primepi(n) + 1 + n
    while m != k:
        m, k = k, primepi(k) + 1 + n
    return m # Chai Wah Wu, Jul 15 2015, updated Apr 14 2016

from sympy import isprime
def ok(n): return n > 1 and not isprime(n)
print([k for k in range(89) if ok(k)]) # Michael S. Branicky, Nov 07 2021

next_A002808=lambda n: next(n for n in range(n,n*5)if not isprime(n)) # next composite >= n > 0; next_A002808(n)==n <=> iscomposite(n). - M. F. Hasler, Mar 28 2025
is_A002808=lambda n:not isprime(n) and n>1 # where isprime(n) can be replaced with: all(n%d for d in range(2, int(n**.5)+1))
# generators of composite numbers:
A002808_upto=lambda stop=1<<59: filter(is_A002808, range(2,stop))
A002808_seq=lambda:(q:=2)and(n for p in primes if (o:=q)<(q:=p) for n in range(o+1,p)) # with, e.g.: primes=filter(isprime,range(2,1<<59)) # M. F. Hasler, Mar 28 2025

[n for n in (2..250) if not is_prime(n)] # G. C. Greubel, Feb 24 2024

a(n) = pi(a(n)) + 1 + n, where pi is the prime counting function.
a(n) = A136527(n, n).
A000005(a(n)) > 2. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) > 1. - Juri-Stepan Gerasimov, Oct 30 2009
A000203(a(n)) < A007955(a(n)). - Juri-Stepan Gerasimov, Mar 17 2011
A066247(a(n)) = 1. - Reinhard Zumkeller, Feb 05 2012
Sum_{n>=1} 1/a(n)^s = Zeta(s)-1-P(s), where P is prime zeta. - Enrique Pérez Herrero, Aug 08 2012
n + n/log n + n/log^2 n < a(n) < n + n/log n + 3n/log^2 n for n >= 4, see Panaitopol. Bojarincev gives an asymptotic version. - Charles R Greathouse IV, Oct 23 2012
a(n) > n + A000720(n) + 1. - François Huppé, Jan 08 2025

Deleted an incomplete and broken link. - N. J. A. Sloane, Dec 16 2010

A104272 Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

Original entry on oeis.org

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653, 659

Offset: 1

Jonathan Sondow, Feb 27 2005

Referring to his proof of Bertrand's postulate, Ramanujan (1919) states a generalization: "From this we easily deduce that pi(x) - pi(x/2) >= 1, 2, 3, 4, 5, ..., if x >= 2, 11, 17, 29, 41, ..., respectively." Since the a(n) are prime (by their minimality), I call them "Ramanujan primes."
See the additional references and links mentioned in A143227.
2n log 2n < a(n) < 4n log 4n for n >= 1, and prime(2n) < a(n) < prime(4n) if n > 1. Also, a(n) ~ prime(2n) as n -> infinity.
Shanta Laishram has proved that a(n) < prime(3n) for all n >= 1.
a(n) - 3n log 3n is sometimes positive, but negative with increasing frequency as n grows since a(n) ~ 2n log 2n. There should be a constant m such that for n >= m we have a(n) < 3n log 3n.
A good approximation to a(n) = R_n for n in [1..1000] is A162996(n) = round(k*n * (log(k*n)+1)), with k = 2.216 determined empirically from the first 1000 Ramanujan primes, which approximates the {k*n}-th prime number which in turn approximates the n-th Ramanujan prime and where abs(A162996(n) - R_n) < 2 * sqrt(A162996(n)) for n in [1..1000]. Since R_n ~ prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), while A162996(n) ~ prime(k*n) ~ k*n * (log(k*n)+1) ~ k*n * log(k*n), A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2). - Daniel Forgues, Jul 29 2009
Let p_n be the n-th prime. If p_n >= 3 is in the sequence, then all integers (p_n+1)/2, (p_n+3)/2, ..., (p_(n+1)-1)/2 are composite numbers. - Vladimir Shevelev, Aug 12 2009
Denote by q(n) the prime which is the nearest from the right to a(n)/2. Then there exists a prime between a(n) and 2q(n). Converse, generally speaking, is not true, i.e., there exist primes outside the sequence, but possess such property (e.g., 109). - Vladimir Shevelev, Aug 14 2009
The Mathematica program FasterRamanujanPrimeList uses Laishram's result that a(n) < prime(3n).
See sequence A164952 for a generalization we call a Ramanujan k-prime. - Vladimir Shevelev, Sep 01 2009
From Jonathan Sondow, May 22 2010: (Start)
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes < 19000 are the lesser of twin primes.
A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (See the arXiv link for a corrected version of Table 1.)
See Shapiro 2008 for an exposition of Ramanujan's proof of his generalization of Bertrand's postulate. (End)
The (10^n)-th R prime: 2, 97, 1439, 19403, 242057, 2916539, 34072993, 389433437, .... - Robert G. Wilson v, May 07 2011, updated Aug 02 2012
The number of R primes < 10^n: 1, 10, 72, 559, 4459, 36960, 316066, 2760321, .... - Robert G. Wilson v, Aug 02 2012
a(n) = R_n = R_{0.5,n} in "Generalized Ramanujan Primes."
All Ramanujan primes are in A164368. - Vladimir Shevelev, Aug 30 2011
If n tends to infinity, then limsup(a(n) - A080359(n-1)) = infinity; conjecture: also limsup(a(n) - A080359(n)) = infinity (cf. A182366). - Vladimir Shevelev, Apr 27 2012
Or the largest prime x such that the number of primes in (x/2,x] equals n. This equivalent definition underlines an important analogy between Ramanujan and Labos primes (cf. A080359). - Vladimir Shevelev, Apr 29 2012
Research questions on R_n - prime(2n) are at A233739, and on n-Ramanujan primes at A225907. - Jonathan Sondow, Dec 16 2013
The questions on R_n - prime(2n) in A233739 have been answered by Christian Axler in "On generalized Ramanujan primes". - Jonathan Sondow, Feb 13 2014
Srinivasan's Lemma (2014): prime(k-n) < prime(k)/2 if R_n = prime(k) and n > 1. Proof: By the minimality of R_n, the interval (prime(k)/2,prime(k)] contains exactly n primes and so prime(k-n) < prime(k)/2. - Jonathan Sondow, May 10 2014
For some n and k, we see that A168421(k) = a(n) so as to form a chain of primes similar to a Cunningham chain. For example (and the first example), A168421(2) = 7, links a(2) = 11 = A168421(3), links a(3) = 17 = A168421(4), links a(4) = 29 = A168421(6), links a(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p-1. - John W. Nicholson, Feb 22 2015
Extending Sondow's 2010 comments: About 48% of primes < 10^9 are Ramanujan primes. About 76% of the lesser of twin primes < 10^9 are Ramanujan primes. - Dana Jacobsen, Sep 06 2015
Sondow, Nicholson, and Noe's 2011 conjecture that pi(R_{m*n}) <= m*pi(R_n) for m >= 1 and n >= N_m (see A190413, A190414) was proved for n > 10^300 by Shichun Yang and Alain Togbé in 2015. - Jonathan Sondow, Dec 01 2015
Berliner, Dean, Hook, Marr, Mbirika, and McBee (2016) prove in Theorem 18 that the graph K_{m,n} is prime for n >= R_{m-1}-m; see A291465. - Jonathan Sondow, May 21 2017
Okhotin (2012) uses Ramanujan primes to prove Lemma 8 in "Unambiguous finite automata over a unary alphabet." - Jonathan Sondow, May 30 2017
Sepulcre and Vidal (2016) apply Ramanujan primes in Remark 9 of "On the non-isolation of the real projections of the zeros of exponential polynomials." - Jonathan Sondow, May 30 2017
Axler and Leßmann (2017) compute the first k-Ramanujan prime for k >= 1 + epsilon; see A277718, A277719, A290394. - Jonathan Sondow, Jul 30 2017

			a(1) = 2 is Bertrand's postulate: pi(x) - pi(x/2) >= 1 for all x >= 2.
a(2) = 11 because a(2) < 8 log 8 < 17 and pi(n) - pi(n/2) > 1 for n = 16, 15, ..., 11 but pi(10) - pi(5) = 1.
Consider a(9)=71. Then the nearest prime > 71/2 is 37, and between a(9) and 2*37, that is, between 71 and 74, there exists a prime (73). - _Vladimir Shevelev_, Aug 14 2009 [corrected by _Jonathan Sondow_, Jun 17 2013]

Srinivasa Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.
Harold N. Shapiro, Ramanujan's idea, Section 9.3B in Introduction to the Theory of Numbers, Dover, 2008.

T. D. Noe, Table of n, a(n) for n = 1..10000
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and Jonathan Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and Jonathan Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
Christian Axler, Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen, Ph.D. thesis 2013, in German, English summary.
Christian Axler, On generalized Ramanujan primes, arXiv:1401.7179 [math.NT], 2014.
Christian Axler, On generalized Ramanujan primes, Ramanujan J. 39 (1) (2016) 1-30.
Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646.
Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika, and C. McBee, Coprime and prime labelings of graphs, arXiv preprint arXiv:1604.07698 [math.CO], 2016; Journal of Integer Sequences, Vol. 19 (2016), #16.5.8.
Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., 9 (1934), 282-288.
Peter Hegarty, Why should one expect to find long runs of (non)-Ramanujan primes?, arXiv:1201.3847 [math.NT], 2012.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Shanta Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory, 6 (2010), 1869-1873.
Catherine Lee, Minimum coprime graph labelings, arXiv:1907.12670 [math.CO], 2019.
Jaban Meher and M. Ram Murty, Ramanujan's proof of Bertrand's postulate, Amer. Math. Monthly, Vol. 120, No. 7 (2013), pp. 650-653.
Alexander Okhotin, Unambiguous finite automata over a unary alphabet, Inf. Comput., 212 (2012), 15-36.
Murat Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.
PlanetMath, Ramanujan prime
Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Juan Matias Sepulcre and Tomás Vidal, On the non-isolation of the real projections of the zeros of exponential polynomials, J. Math. Anal. Appl., 437 (2016) No. 1, 513-525.
Vladimir Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635. Zentralblatt review.
Jonathan 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 Sondow, Ramanujan Prime, Eric Weisstein's MathWorld.
Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, MathWorld.
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 14 (2014), #A19.
Anitha Srinivasan and John W. Nicholson, An improved upper bound for Ramanujan primes, Integers, 15 (2015), #A52.
Wikipedia, Bertrand's postulate.
Wikipedia, Ramanujan prime.
Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., online 14 August 2015, 1-11.

Cf. A006992 (Bertrand primes), A056171 (pi(n) - pi(n/2)).
Cf. A000720, A007053, A014085, A060715, A084139, A084140, A143223, A143224, A143225, A143226, A143227, A080360, A080359, A164368, A164288, A164554, A164333, A164294, A164371, A190303.
Cf. A162996 (Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan Prime).
Cf. A163160 (Round(kn * (log(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime).
Cf. A178127 (Lesser of twin Ramanujan primes), A178128 (Lesser of twin primes if it is a Ramanujan prime).
Cf. A181671 (number of Ramanujan primes less than 10^n).
Cf. A174635 (non-Ramanujan primes), A174602, A174641 (runs of Ramanujan and non-Ramanujan primes).
Cf. A189993, A189994 (lengths of longest runs).
Cf. A190124 (constant of summation: 1/a(n)^2).
Cf. A192820 (2- or derived Ramanujan primes R'_n), A192821, A192822, A192823, A192824, A225907.
Cf. A193761 (0.25-Ramanujan primes), A193880 (0.75-Ramanujan primes).
Cf. A190413, A190414, A212493, A212541, A233739, A233822, A277718, A277719, A164952, A290394, A291465.
Cf. A185004 - A185007 ("modular" Ramanujan primes).
Not to be confused with the Ramanujan numbers or Ramanujan tau function, A000594.

A104272 := proc(n::integer)
    local R;
    if n = 1 then
        return 2;
    end if;
    R := ithprime(3*n-1) ; # upper limit Laishram's thrm Thrm 3 arXiv:1105.2249
    while true do
        if A056171(R) = n then # Defn. 1. of Shevelev JIS 14 (2012) 12.1.1
            return R ;
        end if;
        R := prevprime(R) ;
    end do:
end proc:
seq(A104272(n),n=1..200) ; # slow downstream search <= p(3n-1) R. J. Mathar, Sep 21 2017

(RamanujanPrimeList[n_] := With[{T=Table[{k,PrimePi[k]-PrimePi[k/2]}, {k,Ceiling[N[4*n*Log[4*n]]]}]}, Table[1+First[Last[Select[T,Last[ # ]==i-1&]]],{i,1,n}]]; RamanujanPrimeList[54]) (* Jonathan Sondow, Aug 15 2009 *)
(FasterRamanujanPrimeList[n_] := With[{T=Table[{k,PrimePi[k]-PrimePi[k/2]}, {k,Prime[3*n]}]}, Table[1+First[Last[Select[T,Last[ # ]==i-1&]]],{i,1,n}]]; FasterRamanujanPrimeList[54])
nn=1000; R=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[sT. D. Noe, Nov 15 2010 *)

ramanujan_prime_list(n) = {my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(sSatish Bysany, Mar 02 2017

use ntheory ":all"; my $r = ramanujan_primes(1000); say "[@$r]"; # Dana Jacobsen, Sep 06 2015

a(n) = 1 + max{k: pi(k) - pi(k/2) = n - 1}.
a(n) = A080360(n-1) + 1 for n > 1.
a(n) >= A080359(n). - Vladimir Shevelev, Aug 20 2009
A193761(n) <= a(n) <= A193880(n).
a(n) = 2*A084140(n) - 1, for n > 1. - Jonathan Sondow, Dec 21 2012
a(n) = prime(2n) + A233739(n) = (A233822(n) + a(n+1))/2. - Jonathan Sondow, Dec 16 2013
a(n) = max{prime p: pi(p) - pi(p/2) = n} (see Shevelev 2012). - Jonathan Sondow, Mar 23 2016
a(n) = A000040(A179196(n)). - R. J. Mathar, Sep 21 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = A190303. - Amiram Eldar, Nov 20 2020

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A084138 a(n) is the number of times n is in sequence A060715, i.e., there are exactly a(n) cases where there are exactly n primes between m and 2m, exclusively, for m > 0.

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A182392 Numbers n for which there exists only composite number k such that A060715(k) = n and 2*k-1 is prime, but A104272(n) differs from A080359(n).

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A377338 Numbers k such that A060715(k), the number of prime numbers between k and 2k exclusive, divides k.

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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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A002808 The composite numbers: numbers n of the form x*y for x > 1 and y > 1.

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