%I A000040 M0652 N0241
%S A000040 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A000040 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
%U A000040 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N A000040 The prime numbers.
%C A000040 A number n is prime if it is greater than 1 and has no positive divisors
except 1 and n.
%C A000040 A number n is prime if and only if it has exactly two positive divisors.
%C A000040 A prime has exactly one proper positive divisor, 1.
%C A000040 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
%C A000040 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."
%C A000040 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.
%C A000040 1 is the empty product (has 0 prime factors) whereas a prime has 1 prime
factor (itself). - Daniel Forgues, Jul 23 2009
%C A000040 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.
%C A000040 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
%C A000040 Second sequence ever computed by electronic computer, on EDSAC, May 9
1949 (see Renwick link). - Russ Cox (rsc(AT)swtch.com), Apr 20 2006
%C A000040 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.
%C A000040 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
%C A000040 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
%C A000040 Equals row sums of triangle A143350 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Aug 10 2008]
%C A000040 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...):
%C A000040 APSO(A000040) = A008347=A007504 - 2*(A077126 repeated)
%C A000040 (A007504-A008347)/2 = A077131 Alternated with A077126
%C A000040 For A007504 there is R. J. Mathar, Table of n, a(n) for n = 1..100000
%C A000040 and for A008347 there is T. D. Noe, Table of n, a(n) for n = 0..2000
(End)
%C A000040 The Greek transliteration of 'Prime Number' is 'Proton Arithmon'. [From
Daniel Forgues (squid(AT)zensearch.com), May 08 2009]
%C A000040 Only two divisors of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 25 2009]
%C A000040 omega(n)=number of perfect partitions of n. [From Juri-Stepan Gerasimov
(2stepan(AT)rambler.ru), Oct 29 2009]
%C A000040 Only one prime divisor of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Nov 10 2009]
%C A000040 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]
%D A000040 M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Ann. of Math. (2)
160 (2004), no. 2, 781-793.
%D A000040 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin,
2nd. ed., 2001; see p. 3.
%D A000040 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 2.
%D A000040 E. Bach and J. O. Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.
%D A000040 P. T. Bateman and H. G. Diamond, A hundred years of prime numbers, Amer.
Math. Monthly, Vol. 103 (1996) pp. 729-741.
%D A000040 D. M. Bressoud, Factorization and Primality Testing, Springer-Verlag
NY 1989.
%D A000040 Michele Cipolla, La determinazione assintotica dell'nimo numero primo,
Matematiche Napoli 3 (1902), 132-166.
%D A000040 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective,
Springer, NY, 2001; see p. 1.
%D A000040 J.-P. Delahaye, Merveilleux nombres premiers, Pour la Science-Belin Paris,
2000.
%D A000040 J.-P. Delahaye, Savoir si un nombre est premier: facile, Pour La Science,
303(1) 2003, pp. 98-102.
%D A000040 M. Dietzfelbinger, Primality Testing in Polynomial Time, Springer NY
2004.
%D A000040 U. Dudley, Formulas for primes, Math. Mag., 56 (1983), 17-22.
%D A000040 Pierre Dusart, Autour de la fonction qui compte le nombre de nombres
premiers, Dissertation, Universite de Limoges (1998).
%D A000040 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.
%D A000040 J. Elie, "L'algorithme AKS", in 'Quadrature', No. 60, pp. 22-32, 2006
EDP-sciences, Les Ulis (France);
%D A000040 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.
%D A000040 W. & F. Ellison, Prime Numbers, Hermann Paris 1985
%D A000040 T. Estermann, Introduction to Modern Prime Number Theory, Camb. Univ.
Press, 1969.
%D A000040 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
3rd ed., Oxford Univ. Press, 1954, p. 2.
%D A000040 H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput.
17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
%D A000040 M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press,
1972.
%D A000040 D. S. Jandu, Prime Numbers And Factorization, Infinite Bandwidth Publishing,
N. Hollywood CA 2007.
%D A000040 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea,
NY, 1974.
%D A000040 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 A000040 D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie
Institute, Washington, D.C. 1909.
%D A000040 W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA,
2 vols., 1956, Vol. 1, Chap. 6.
%D A000040 H. Lifchitz, Table Des nombres Premiers de 0 a 20 millions (Tomes I &
II), Albert Blanchard, Paris 1971.
%D A000040 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
%D A000040 Kaoru Motose, On values of cyclotomic polynomials. II, Math. J. Okayama
Univ. 37 (1995), 27-36.
%D A000040 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY
1995.
%D A000040 P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.
%D A000040 H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhaeuser
Boston, Cambridge MA 1994.
%D A000040 B. Rittaud, "31415879. Ce nombre est-il premier?" ['Is this number prime?'],
La Recherche, Vol. 361, pp. 70-73, Feb 15 2003, Paris.
%D A000040 J. Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers,
American Journal of Mathematics 63 (1941) 211-232.
%D A000040 M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins,
2003; see p. 5.
%D A000040 D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea,
1978, Chap. 1.
%D A000040 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000040 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000040 D. Wells, Prime Numbers:The Most Mysterious Figures In Math, J.Wiley
NY 2005.
%D A000040 Wikipedia, Prime Number Theorem.
%D A000040 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
%H A000040 N. J. A. Sloane, <a href="b000040.txt">Table of n, prime(n) for n = 1..10000</
a>
%H A000040 N. J. A. Sloane, <a href="a000040.txt">Table of n, prime(n) for n = 1..100000</
a>
%H A000040 M. Agrawal, N. Kayal & N. Saxena, PRIMES is in P, <a href="http://www.cse.iitk.ac.in/
users/manindra/primality_original.pdf">Original Preprint</a>; <a
href="http://www.cse.iitk.ac.in/users/manindra/primality_v6.pdf">
September 2005 Version</a>
%H A000040 M. Agrawal, N. Kayal & N. Saxena, <a href="http://projecteuclid.org/Dienst/
UI/1.0/Summarize/euclid.annm/1111770735">PRIMES is in P</a>, Annals
of Maths., 160 no.2 (2004) pp. 781-793
%H A000040 P. Alfeld, <a href="http://www.math.utah.edu/~alfeld/math/prime.html">
Notes and Literature on Prime Numbers</a>
%H A000040 Anonymous, <a href="http://e.co.za/primes.html">Prime Numbers (Applet)</
a>
%H A000040 Anonymous, <a href="http://www.mathematical.com/primelist1to100kk.html">
Prime Number Master Index (for primes up to 2*10^7)</a>
%H A000040 Anonymous, <a href="http://www.primzahlen.de/files/theorie/liste.htm">
Primzahlenliste(Prime List Generator)</a>
%H A000040 Anonymous, <a href="http://everything2.net/index.pl?node_id=74889&displaytype=printable&lastnode_id=74889">
prime number</a>
%H A000040 D. J. Bernstein, <a href="http://cr.yp.to/papers/aks.pdf">Proving Primality
After Agrawal-Kayal-Saxena</a>
%H A000040 D. J. Bernstein, <a href="http://cr.yp.to/primetests.html">Distinguishing
prime numbers from composite numbers</a>
%H A000040 P. Berrizbeitia, <a href="http://arXiv.org/abs/math.NT/0211334">Sharpening
"Primes is in P" for a large family of numbers</a>
%H A000040 A. Booker, <a href="http://primes.utm.edu/nthprime">The Nth Prime Page</
a>
%H A000040 F. Bornemann, <a href="http://www.ams.org/notices/200305/fea-bornemann.pdf">
PRIMES Is in P:A Breakthrough for "Everyman"</a>
%H A000040 A. Bowyer, <a href="http://www.bath.ac.uk/~ensab/Primes">Formulae for
Primes</a>
%H A000040 B. M. Bredikhin, <a href="http://eom.springer.de/P/p074530.htm">Prime
number</a>
%H A000040 J. Brennan, <a href="http://jamesbrennan.org/algebra/prime_list.html">
Prime Number List Server</a>
%H A000040 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub120.html">
Primality testing and integer factorization</a>
%H A000040 J. Britton, <a href="http://britton.disted.camosun.bc.ca/jbprimelist.htm">
Prime Number List</a>
%H A000040 D. Butler, <a href="http://www.tsm-resources.com/alists/prim.html">The
first 2000 Prime Numbers</a>
%H A000040 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/">The Prime
Pages</a>
%H A000040 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=TablesOfPrimes">
Tables of primes</a>
%H A000040 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/small/
10000.txt">The first 10000 primes</a>
%H A000040 C. K. Caldwell, <a href="http://primes.utm.edu/curios/includes/file.php?file=primetest.html">
A Primality Test</a>
%H A000040 M. Chamness, <a href="http://www.alumni.caltech.edu/~chamness/prime.html">
Prime number generator (Applet)</a>
%H A000040 J.-L. Cooke, <a href="http://www.jlcooke.ca/Numbers/Primes.shtml">Prime
Numbers(Primality Tester)</a>
%H A000040 P. Cox, <a href="http://members.cox.net/mathmistakes/primes.htm">Primes
is in P</a>
%H A000040 P. J. Davis & R. Hersh, The Mathematical Experience, <a href="http://
www.fortunecity.com/emachines/e11/86/mathex5.html">The Prime Number
Theorem</a>
%H A000040 J.-M. De Koninck, <a href="http://campmath.uqam.ca/infos2004/conf_double.pdf">
Les nombres premiers: mysteres et consolation</a>
%H A000040 J.-P. Delahaye, <a href="http://www.cnrs.fr/Cnrspresse/math2000/html/
math10.htm">Formules et nombres premiers</a>
%H A000040 Desmatron, <a href="http://desmatron.altervista.org/desmath/desprimes.html">
Primes 2 through 101477</a>
%H A000040 J. Elie, <a href="http://www.trigofacile.com/maths/curiosite/primarite/
aks/pdf/algorithme-aks.pdf">L'algorithme AKS ou Les nombres premiers
sont de classe P</a>
%H A000040 C. P. Estany, <a href="http://www.geocities.com/CapeCanaveral/Launchpad/
2208/PRIMERS.TXT">List of (148933) Prime Numbers 1 through 2000000</
a>
%H A000040 L. Euler, <a href="http://arXiv.org/abs/math.HO/0501118">Observations
on a theorem of Fermat and others on looking at prime numbers</a>
%H A000040 W. Fendt, <a href="http://www.walter-fendt.de/m14e/primes.htm">Table
of Primes from 1 to 1000000000000</a>
%H A000040 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/
05011379">On the non-holonomic character of logarithms, powers and
the n-th prime function</a>
%H A000040 J. Flamant, <a href="http://jocelyn.smoofy.net/np/cache/index.html">Primes
up to one million</a>
%H A000040 K. Ford, Expositions of the <a href="http://www.math.uiuc.edu/~ford">
PRIMES is in P</a> theorem.
%H A000040 L. & Y. Gallot, <a href="http://perso.wanadoo.fr/yves.gallot/primes/chrrcds.html">
The Chronology of Prime Number Records</a>
%H A000040 P. Garrett, <a href="http://www.math.umn.edu/~garrett/crypto/overheads01.pdf">
Big Primes, Factoring Big Integers</a>
%H A000040 P. Garrett, <a href="http://www.math.umn.edu/~garrett/js/naive_prim_test.html">
Naive Primality Test</a>
%H A000040 P. Garrett, <a href="http://www.math.umn.edu/~garrett/js/list_pr.html">
Listing Primes</a>
%H A000040 N. Gast, <a href="http://www.eleves.ens.fr/home/gast/misc/GastCrypto.pdf">
PRIMES is in P: Manindra Agrawal, Neeraj Kayal and Nitin Saxena</
a>
%H A000040 D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, <a href="http:/
/arXiv.org/abs/math.NT/0506067">Small gaps between primes and almost
primes</a>
%H A000040 A. Granville, <a href="http://math.stanford.edu/~brubaker/granville.pdf">
It Is Easy To Determine Whether A Given Number Is Prime</a>
%H A000040 A. Granville, <a href="http://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/
home.html">It is easy to determine whether a given integer is prime</
a>
%H A000040 P. Hartmann, <a href="http://translate.google.com/translate?hl=en&sl=de&u=http:/
/www.beweise.mathematic.de">Prime number proofs</a>
%H A000040 ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/primes.htm">
List of first 50000 primes grouped within ten columns</a>
%H A000040 N. Kayal & N. Saxena, Resonance 11-2002, <a href="http://www.ias.ac.in/
resonance/Nov2002/pdf/Nov2002ResearchNews.pdf">A polynomial time
algorithm to test if a number is prime or not</a>
%H A000040 M.-H. Kim, <a href="http://www.math.snu.ac.kr/~mhkim/t-unsol.pdf">Unsolved
Problems In Number Theory</a>
%H A000040 J.-M. De Koninck, <a href="http://campmath.uqam.ca/2005/nbPremMysEnj.pdf">
Nombres premiers: mysteres et enjeux</a>
%H A000040 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A000040 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A000040 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http:/
/www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
c=umhistmath;idno=ABV2766.0001.001">vol. 1</a> and <a href="http:/
/www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
c=umhistmath;idno=ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin,
B. G. Teubner, 1909.
%H A000040 D. N. Lehmer, <a href="http://www.fortunecity.com/emachines/e11/86/graphics/
mathex/PRIMES.gif">Table of the First 2500 Prime Numbers</a>, Carnegie
Institute of Washington,1914.
%H A000040 W. Liang & H. Yan, <a href="http://fr.arXiv.org/abs/math.NT/0603450">
Pseudo Random test of prime numbers</a>
%H A000040 J. Malkevitch, <a href="http://www.ams.org/featurecolumn/archive/primes1.html">
Primes</a>
%H A000040 MathIsFun.com, <a href="http://www.mathisfun.com/prime_numbers.html">
Prime Numbers Chart</a>
%H A000040 Mathworld Headline News, <a href="http://mathworld.wolfram.com/news/2002-08-07/
primetest/">Primality Testing is Easy</a>
%H A000040 K. Matthews, <a href="http://www.numbertheory.org/php/prime_generator.html">
Generating prime numbers</a>
%H A000040 Y. Motohashi, <a href="http://arXiv.org/abs/math.HO/0512143">Prime numbers-your
gems</a>
%H A000040 J. Moyer, <a href="http://www.rsok.com/~jrm/printprimes.html">Some Prime
Numbers</a>
%H A000040 C. W. Neville, <a href="http://arXiv.org/abs/math.NT/0210282">New Results
on Primes from an Old Proof of Euler's</a>
%H A000040 L. C. Noll, <a href="http://www.isthe.com/chongo/tech/math/prime/index.html">
Prime numbers, Mersenne Primes, Perfect Numbers, etc.</a>
%H A000040 J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/
~history/HistTopics/Prime_numbers.html">Prime Numbers</a>
%H A000040 M. Ogihara & S. Radziszowski, <a href="http://www.cs.rit.edu/~spr/PUBL/
primes.pdf">Agrawal-Kayal-Saxena Algorithm for Testing Primality
in Polynomial Time</a>
%H A000040 J. M. Parganin, <a href="http://noe-education.org/D11102.php">Primes
less than 50000</a>
%H A000040 K. Peavey, <a href="http://www.geocities.com/ResearchTriangle/Thinktank/
2434/prime/primenumbers.html">Prime List Display in batches of 50000</
a>
%H A000040 I. Peterson, <a href="http://www.fortunecity.com/emachines/e11/86/tourist2b.html">
Prime Pursuits</a>
%H A000040 O. E. Pol, <a href="http://www.polprimos.com">Numeros primos</a>
%H A000040 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/4.jpg">Illustration
of initial terms</a>.
%H A000040 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polprdipi.jpg">
Divisors and pi(x)</a>
%H A000040 Prime-Numbers.org, <a href="http://www.prime-numbers.org">Prime-Numbers.org(Prime
Tester & List Server)</a>
%H A000040 Primefan, <a href="http://primefan.tripod.com/500Primes1.html">The First
500 Prime Numbers</a>
%H A000040 Primefan, <a href="http://primefan.tripod.com/PrimeLister.html">Script
to Calculate Prime Numbers</a>
%H A000040 Project Gutenberg Etext, <a href="http://www.gutenberg.org/etext/65">
First 100,000 Prime Numbers</a>
%H A000040 C. D. Pruitt, <a href="http://www.mathematical.com/mathprimegen.html">
Formulae for Generating All Prime Numbers</a>
%H A000040 R. Ramachandran, Frontline 19 (17) 08-2000, <a href="http://www.flonnet.com/
fl1917/19171290.htm">A Prime Solution</a>
%H A000040 W. S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC
log</a>.
%H A000040 F. Richman, <a href="http://www.math.fau.edu/Richman/primes.htm">Generating
primes by the sieve of Eratosthenes</a>
%H A000040 J. Barkley Rosser and Lowell Schoenfeld, <a href="a000720.html">Approximate
formulas for some functions of prime numbers</a> (scan of some key
pages from an ancient annotated photocopy)
%H A000040 S. M. Ruiz and J. Sondow, <a href="http://arXiv.org/abs/math.NT/0210312">
Formulas for pi(n) and the n-th prime</a>.
%H A000040 S. O. S. Math, <a href="http://www.sosmath.com/tables/prime/prime.html">
First 1000 Prime Numbers</a>
%H A000040 A. Schulman, <a href="http://www.sonic.net/~undoc/java/PrimeCalc.html">
Prime Number Calculator</a>
%H A000040 M. Slone, PlanetMath.Org, <a href="http://planetmath.org/encyclopedia/
FirstThousandPositivePrimeNumbers.html">First thousand positive prime
numbers</a>
%H A000040 A. Stiglic, <a href="http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html">
The PRIMES is in P little FAQ</a>
%H A000040 S. Stepney, <a href="http://public.logica.com/~stepneys/cyc/p/prime100.htm">
Primes 2 through 10000</a>
%H A000040 Tomas Svoboda, <a href="http://www.svobodat.com/primes/PRIMES1T.TXT">
List of primes up to 10^6</a> [Slow link] (From R. J. Mathar, Jul
23 2009)
%H A000040 J. Teitelbaum, <a href="http://www.ams.org/bull/2002-39-03/S0273-0979-02-00947-3/
S0273-0979-02-00947-3.pdf">Review of "Prime numbers:A computational
perspective" by R.Crandall & C.Pomerance</a>
%H A000040 K. Thomas, <a href="http://students.bath.ac.uk/ma1ktt/Primes/primes_home.html">
Prime Numbers</a>
%H A000040 J. Thonnard, <a href="http://www.proftnj.com/calcprem.htm">Les nombres
premiers(Primality check; Closest next prime; Factorizer)</a>
%H A000040 A. Turpel, <a href="http://www2.vo.lu/homepages/armand/index.html">Aesthetics
of the Prime Sequence</a>
%H A000040 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Premier/introduc.htm">Nombres Premiers</a>
%H A000040 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/
Wwwgvmm/Premier/DiMille.htm">Primes up to 10000</a>
%H A000040 S. Wagon, <a href="http://www.americanscientist.org/template/BookReviewTypeDetail/
assetid/14442?&print=yes">Prime Time : Review of "Prime Numbers:A
Computational Perspective" by R. Crandall & C. Pomerance</a>
%H A000040 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/unusual.htm">
unusual and physical methods for finding prime numbers</a>
%H A000040 S. Wedeniwski, <a href="http://w210.Ub.uni-tuebingen.de/dbt/volltexte/
2001/420/pdf/dissertation.pdf">Primality Tests on Commutator Curves</
a>
%H A000040 E. Wegrzynowski, <a href="http://www.lifl.fr/~wegrzyno/FormulPrem/FormulesPremiers23.html">
Les formules simples qui donnent des nombres premiers en grande quantites</
a>
%H A000040 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PrimeNumber.html">Link to a section of The World of Mathematics (1).</
a>
%H A000040 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PrimePower.html">Link to a section of The World of Mathematics (2).</
a>
%H A000040 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
AlmostPrime.html">Link to a section of The World of Mathematics (3).</
a>
%H A000040 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Prime-GeneratingPolynomial.html">Link to a section of The World of
Mathematics (4)</a>
%H A000040 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PrimeSpiral.html">Link to a section of The World of Mathematics (5)</
a>
%H A000040 Wikipedia, <a href="http://www.wikipedia.org/wiki/Prime_number">Prime
number</a>
%H A000040 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Primes.html">
Prime Generator(between two bounds)</a>
%H A000040 G. Xiao, Primes server, <a href="http://wims.unice.fr/~wims/en_tool~number~primes.html">
Sequential Batches Primes Listing (up to orders not exceeding 10^308)</
a>
%H A000040 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">
To display p(n) for n up to 41561, operate on "prime(n)"</a>
%H A000040 Z. Zheng, <a href="http://www.linguistlist.org/~zheng/courseware/showprime.html">
"Show Prime Numbers" server [p(n),n=1 up to 10^10]</a> [Broken link?]
%H A000040 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000040 The prime number theorem is the statement that a(n) ~ n * log n as n
-> infinity (Hardy and Wright, page 10).
%F A000040 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]
%F A000040 For all n, a(n) > n log n. [Rosser]
%F A000040 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]
%F A000040 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]
%F A000040 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
%F A000040 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
%F A000040 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]
%F A000040 A000005(a(n))=2; A002033(a(n+1))=1 [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 17 2009]
%F A000040 A001222(a(n))=1. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Nov 10 2009]
%p A000040 A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
%t A000040 Table[ Prime[n], {n, 1, 60} ]
%o A000040 (MAGMA) [ n : n in [2..500] | IsPrime(n) ];
%o A000040 (MAGMA) a := func< n | NthPrime(n) >;
%o A000040 (PARI) a(n)=if(n<1,0,prime(n))
%o A000040 # (SAGE) Demonstration program from Jaap Spies:
%o A000040 # To see which functions are available type: sloane.A[tab]
%o A000040 # All builtin SAGE programs are called the same way:
%o A000040 # a = sloane.A000040; a # This returns the name of the sequence
%o A000040 # a(n) # This returns the n-th number of the sequence:
%o A000040 # a.list(n) # This returns a list of the first n numbers:
%o A000040 # Copy and paste the following into a worksheet or the interpreter:
%o A000040 a = sloane.A000040; print a
%o A000040 print a(1)
%o A000040 print a(2)
%o A000040 print a(58)
%o A000040 print a.list(58)
%o A000040 (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]
%o A000040 (Other) sage: prime_range(1,300) # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 27 2009]
%Y A000040 Cf. A000027, A018252, A002808, A008578, A006879, A006880.
%Y A000040 Cf. also A000720 ("pi"), A001223 (differences between primes), A001358
("semiprimes").
%Y A000040 A143350 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2008]
%Y A000040 Cf. A000005, A001221, A002033. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Oct 29 2009]
%Y A000040 Sequence in context: A055398 A070159 A158611 this_sequence A008578 A100726
A015919
%Y A000040 Adjacent sequences: A000037 A000038 A000039 this_sequence A000041 A000042
A000043
%K A000040 core,nonn,nice,easy,new
%O A000040 1,1
%A A000040 N. J. A. Sloane (njas(AT)research.att.com).
%E A000040 Additional links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com),
Dec 23 2003
%E A000040 Additional comments from Jonathan Sondow (jsondow(AT)alumni.princeton.edu),
Dec 27 2004
%E A000040 Updated geocities.com links - R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Oct 30 2009
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