1,1

A number n is prime if it is greater than 1 and has no positive divisors except 1 and n.

A number n is prime if and only if it has exactly two positive divisors.

A prime has exactly one proper positive divisor, 1.

The sum of an odd number > 1 (2i+1, i >= 1) of consecutive positive odd numbers centered on the jth odd number >= 2i+1 (2j+1, j >= i) being (2i+1)*(2j+1) has 2 or more odd prime factors (odd semiprime iff 2i+1 and 2j+1 are primes). - Daniel Forgues (squid(AT)zensearch.com), Jul 15 2009

Comment from Pieter Moree, Oct 14 2004: 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."

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.

1 is the empty product (has 0 prime factors) whereas a prime has 1 prime factor (itself). - Daniel Forgues, Jul 23 2009

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.

Elementary primality test: If no prime =<sqrt(m) divides m, then m is prime.(since a prime is its own exclusive multiple, apart from 1) - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 31 2005

Second sequence ever computed by electronic computer, on EDSAC, May 9 1949 (see Renwick link). - Russ Cox (rsc(AT)swtch.com), Apr 20 2006

Every prime p is a linear combination of previous primes p(n) with nonzero coefficients c(n) and |c(n)| < p(n). - Amarnath Murthy, Franklin T. Adams-Watters and Joshau Zucker, May 17 2006.

Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n=n, for k=1). See A111774. - Jaap Spies (j.spies(AT)hccnet.nl), Jan 04 2007

There is a unique decomposition of the primes: provided the weight A117078(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) + A001223(n). - Remi Eismann (reismann(AT)free.fr), Feb 16 2007

Equals row sums of triangle A143350 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 28 2008: (Start). APSO (Alternating partial sums of sequence) a-b+c-d+e-f+g...=(a+b+c+d+e+f+g...)-2*(b+d+f...):

APSO(A000040) = A008347=A007504 - 2*(A077126 repeated)

(A007504-A008347)/2 = A077131 alternated with A077126. (End)

The Greek transliteration of 'Prime Number' is 'Proton Arithmon'. [From Daniel Forgues (squid(AT)zensearch.com), May 08 2009]

omega(n)=number of perfect partitions of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 29 2009]

Numbers with exactly one prime divisor. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 10 2009]

2*number of divisors of n*n=3*number of perfect partitions of n*n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler), Nov 19 2009]

Union of trivial (2 and 3) and nontivial (A168546) primes. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 30 2009]

M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781-793.

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 J. O. Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.

P. T. Bateman and H. G. Diamond, A hundred years of prime numbers, Amer. Math. Monthly, Vol. 103 (1996) pp. 729-741.

D. M. Bressoud, Factorization and Primality Testing, Springer-Verlag NY 1989.

Michele Cipolla, La determinazione assintotica dell'nimo numero primo, Matematiche Napoli 3 (1902), 132-166.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 1.

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.

U. Dudley, Formulas for primes, Math. Mag., 56 (1983), 17-22.

Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Dissertation, Universite de Limoges (1998).

Pierre Dusart, The kth 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", in 'Quadrature', No. 60, pp. 22-32, 2006 EDP-sciences, Les Ulis (France);

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.

W. & F. Ellison, Prime Numbers, Hermann Paris 1985

T. Estermann, Introduction to Modern Prime Number Theory, Camb. Univ. Press, 1969.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035

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 a 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

Kaoru Motose, On values of cyclotomic polynomials. II, Math. J. Okayama Univ. 37 (1995), 27-36.

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, Birkhaeuser 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.

J. Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers, American Journal of Mathematics 63 (1941) 211-232.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 5.

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).

D. Wells, Prime Numbers:The Most Mysterious Figures In Math, J.Wiley NY 2005.

Wikipedia, Prime Number Theorem.

H. C. Williams and J. O. 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, Original Preprint; September 2005 Version

M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, Annals of Maths., 160 no.2 (2004) pp. 781-793

P. Alfeld, Notes and Literature on Prime Numbers

Anonymous, Prime Numbers (Applet)

Anonymous, Prime Number Master Index (for primes up to 2*10^7)

Anonymous, Primzahlenliste(Prime List Generator)

Anonymous, prime number

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

A. Booker, The Nth Prime Page

F. Bornemann, PRIMES Is in P:A Breakthrough for "Everyman"

A. Bowyer, Formulae for Primes

B. M. Bredikhin, Prime number

J. Brennan, Prime Number List Server

R. P. Brent, Primality testing and integer factorization

J. Britton, Prime Number List

D. Butler, The first 2000 Prime Numbers

C. K. Caldwell, The Prime Pages

C. K. Caldwell, Tables of primes

C. K. Caldwell, The first 10000 primes

C. K. Caldwell, A Primality Test

M. Chamness, Prime number generator (Applet)

J.-L. Cooke, Prime Numbers(Primality Tester)

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: mysteres et consolation

J.-P. Delahaye, Formules et nombres premiers

Desmatron, Primes 2 through 101477

J. Elie, L'algorithme AKS ou Les nombres premiers sont de classe P

C. P. Estany, List of (148933) Prime Numbers 1 through 2000000

L. Euler, Observations on a theorem of Fermat and others on looking at prime numbers

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

J. Flamant, Primes up to one million

K. Ford, Expositions of the PRIMES is in P theorem.

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

D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between primes and almost primes

A. Granville, It Is Easy To Determine Whether A Given Number Is Prime

A. Granville, It is easy to determine whether a given integer is prime

P. Hartmann, Prime number proofs

ICON Project, List of first 50000 primes grouped within ten columns

N. Kayal & N. Saxena, Resonance 11-2002, A polynomial time algorithm to test if a number is prime or not

M.-H. Kim, Unsolved Problems In Number Theory

J.-M. De Koninck, Nombres premiers: mysteres et enjeux

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

D. N. Lehmer, Table of the First 2500 Prime Numbers, Carnegie Institute of Washington,1914.

W. Liang & H. Yan, Pseudo Random test of prime numbers

J. Malkevitch, Primes

MathIsFun.com, Prime Numbers Chart

Mathworld Headline News, Primality Testing is Easy

K. Matthews, Generating prime numbers

Y. Motohashi, Prime numbers-your gems

J. Moyer, Some Prime Numbers

C. W. Neville, New Results on Primes from an Old Proof of Euler's

L. C. Noll, Prime numbers, Mersenne Primes, Perfect Numbers, etc.

J. J. O'Connor & E. F. Robertson, Prime Numbers

M. Ogihara & S. Radziszowski, Agrawal-Kayal-Saxena Algorithm for Testing Primality in Polynomial Time

J. M. Parganin, Primes less than 50000

K. Peavey, Prime List Display in batches of 50000

I. Peterson, Prime Pursuits

O. E. Pol, Numeros primos

O. E. Pol, Illustration of initial terms.

O. E. Pol, Divisors and pi(x)

Prime-Numbers.org, Prime-Numbers.org(Prime Tester & List Server)

Primefan, The First 500 Prime Numbers

Primefan, 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

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy)

S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime.

S. O. S. Math, First 1000 Prime Numbers

A. Schulman, Prime Number Calculator

M. Slone, PlanetMath.Org, First thousand positive prime numbers

A. Stiglic, The PRIMES is in P little FAQ

S. Stepney, Primes 2 through 10000

Tomas Svoboda, List of primes up to 10^6 [Slow link] (From R. J. Mathar, Jul 23 2009)

J. Teitelbaum, Review of "Prime numbers:A computational perspective" by R.Crandall & C.Pomerance

K. Thomas, Prime Numbers

J. Thonnard, Les nombres premiers(Primality check; Closest next prime; Factorizer)

A. Turpel, Aesthetics of the Prime Sequence

G. Villemin's Almanach of Numbers, Nombres Premiers

G. Villemin's Almanac of Numbers, Primes up to 10000

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 quantites

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (4)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (5)

Wikipedia, Prime number

D. Williams, Prime Generator(between two bounds)

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)"

Z. Zheng, "Show Prime Numbers" server [p(n),n=1 up to 10^10] [Broken link?]

Index entries for "core" sequences

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 2 - 2) + O( n (log log n)^2/ (log n)^2). [Cipoli, quoted in the Wikipedia article]

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 (jsondow(AT)alumni.princeton.edu), Mar 06 2004

I conjecture that Sum(1/(p(i)*log(p(i)))=Pi/2=1.570796327... Sum(1/(i=1..100000 p(i)*log(p(i)))=1.565585514... It converges very slowly. - Miklos Kristof (kristmikl(AT)freemail.hu), Feb 12 2007

The last conjecture has been discussed by the math.research newsgroup recently. The sum, which is greater than pi/2, is computed by Mathar in sequence A137245. [From T. D. Noe (noe(AT)sspectra.com), Jan 13 2009]

A000005(a(n))=2; A002033(a(n+1))=1 [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 17 2009]

A001222(a(n))=1. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 10 2009]

A000040 := n->ithprime(n); [ seq(ithprime(i), i=1..100) ];

Table[ Prime[n], {n, 1, 60} ]

(MAGMA) [ n : n in [2..500] | IsPrime(n) ];

(MAGMA) a := func< n | NthPrime(n) >;

(PARI) a(n)=if(n<1, 0, prime(n))

# (SAGE) Demonstration program from Jaap Spies:

# To see which functions are available type: sloane.A[tab]

# All builtin SAGE programs are called the same way:

# a = sloane.A000040; a # This returns the name of the sequence

# a(n) # This returns the n-th number of the sequence:

# a.list(n) # This returns a list of the first n numbers:

# Copy and paste the following into a worksheet or the interpreter:

a = sloane.A000040; print a

print a(1)

print a(2)

print a(58)

print a.list(58)

(PARI) The program below is supposedly valid for generating primes for n>=3; it is based on the comment in A075888: "For n>=3, prime(n+1)^2-prime(n)^2 is always divisible by 24" j=[]; for(n=0, 500, if((floor(sqrt(4!*(n+1) + 1))) == ceil(sqrt(4!*(n+1) + 1)), if(isprime(floor(sqrt(4!*(n+1) + 1))), j=concat(j, floor(sqrt(4!*(n+1) + 1)))))); j [From Alexander R. Povolotsky (pevnev(AT)juno.com), Sep 16 2008]

(Other) sage: prime_range(1, 300) # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 27 2009]

Cf. A000027, A018252, A002808, A008578, A006879, A006880.

Cf. also A000720 ("pi"), A001223 (differences between primes), A001358 ("semiprimes").

A143350 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]

Cf. A000005, A001221, A002033. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 29 2009]

Sequence in context: A055398 A070159 A158611 this_sequence A008578 A100726 A015919

Adjacent sequences: A000037 A000038 A000039 this_sequence A000041 A000042 A000043

core,nonn,nice,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 23 2003

Additional comments from Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 27 2004

Updated geocities.com links - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009

Removed duplicate of one comment and material non associated with this sequence - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 16 2009