%I A000720 M0256 N0090
%S A000720 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,
%T A000720 11,11,11,11,12,12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,
%U A000720 16,16,16,17,17,18,18,18,18,18,18,19,19,19,19,20,20,21,21,21,21,21,21
%N A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish
it from the number 3.14159...
%C A000720 Partial sums of A010051 (characteristic function of primes). - Jeremy
Gardiner (jeremy.gardiner(AT)btinternet.com), Aug 13 2002
%C A000720 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 (jsondow(AT)alumni.princeton.edu),
Dec 27 2004
%C A000720 The g.f. -z*(-1-z-z**3-z**5+z**6+z**7)/((1+z)*(z**2-z+1)*(z**2+z+1)*(z-1)**2)
conjectured by S. Plouffe in his 1992 dissertation is wrong.
%C A000720 See the additional references and links mentioned in A143227. [From Jonathan
Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
%C A000720 Equals row sums of triangle A143538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Aug 23 2008]
%C A000720 a(n) = A036234(n)-1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Mar 23 2009]
%C A000720 ((10^n)^2)/(ln((10^n)!)) [From Eric Desbiaux (moongerms(AT)wanadoo.fr),
Jul 15 2009]
%D A000720 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.
%D A000720 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 8.
%D A000720 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math.
Soc., 1963; p. 129.
%D A000720 P. T. Bateman & H. G. Diamond, "A Hundred Years of Prime Numbers", Amer.
Math. Month., Vol. 103 (9), Nov. 1996, pp. 729-741, MAA Washington
DC.
%D A000720 Bressoud & Wagon, A Course in Computational Number Theory, Springer/Key,
2000 (with a Mathematica package for computational number theory);
wagon_notes.nb: http://www.msri.org/publications/ln/msri/2000/introant/
wagon/mma/wagon_notes.nb
%D A000720 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective,
Springer, NY, 2001; see p. 5.
%D A000720 Pierre Dusart, Autour de la fonction qui compte le nombre de nombres
premiers, Dissertation, Universite de Limoges (1998).
%D A000720 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 A000720 R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation
monoid, Discrete Math., 308 (2008), 4801-4810.
%D A000720 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,
5th ed., Oxford Univ. Press, 1979, Theorems 6, 7, 420.
%D A000720 G. J. O. Jameson, The Prime Number Theorem, Camb.Univ.Press 2003
%D A000720 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.1.
(For inequalities, etc.)
%D A000720 W. Narkiewicz, The Development of Prime Number Theory, Springer-Verlag
2000.
%D A000720 J. Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers,
American Journal of Mathematics 63 (1941) 211-232.
%D A000720 J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some
functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.
%D A000720 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000720 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000720 G. Tenebaum and M. Mendes France, Prime Numbers and Their Distribution,
AMS Providence RI 1999
%D A000720 Wikipedia, Prime Number Theorem.
%H A000720 Daniel Forgues, <a href="b000720.txt">Table of n, pi(n) for n = 1..100000</
a>
%H A000720 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000720 S. Bennett, <a href="http://www.freewebs.com/history_of_mathematics">
The role of Riemann's zeta function in the analytic proof of the
Prime Number Theorem</a>
%H A000720 C. Bonanno & M. S. Mega, <a href="http://arXiv.org/abs/cond-mat/0309251">
Toward a dynamic model for prime numbers</a>
%H A000720 D. M. Bressoud, <a href="http://www.maa.org/reviews/primenumbertheorem.html">
Review of "The Prime Number Theorem" by G. J. O. Jameson</a>
%H A000720 B. Brubaker, <a href="http://math.stanford.edu/~brubaker/pnt.pdf">The
Prime Number Theorem</a>
%H A000720 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
page.php/PrimeNumberThm.html">Prime number theorem</a>
%H A000720 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml">
How Many Primes Are There</a>
%H A000720 W. W. L. Chen, <a href="http://www.maths.mq.edu.au/~wchen/lndpnfolder/
lndpn.html">Distribution of Prime Numbers</a>
%H A000720 M. Deleglise, <a href="http://algo.inria.fr/dumas/Algosem/deleglise9596/
">Computation of large values of pi(x)</a>
%H A000720 Encyclopedia Britannica, <a href="http://users.forthnet.gr/ath/kimon/
PNT/Prime%20Number%20Theorem.htm">The Prime Number Theorem</a>
%H A000720 G. H. Hardy & J. E. Littlewood, <a href="http://users.ift.uni.wroc.pl/
~mwolf/Hardy_Littlewood%20zeta.pdf">Contributions To The Theory Of
The Riemann Zeta-Function And The Theory Of The Distribution Of Primes</
a>
%H A000720 M. Hassani, <a href="http://jipam.vu.edu.au/issues.php?op=viewissue&issue=86">
Approximation of pi(x) by Psi(x)</a>, J. Inequ. Pure Appl. Math.
7 (2006) vol. 1, #7
%H A000720 Y.-C. Kim, <a href="http://arXiv.org/abs/math/0502062">Note on the Prime
Number Theorem</a>
%H A000720 T. V. Kolev, <a href="http://citeseer.nj.nec.com/correct/495414">On the
number of Prime Numbers less than a Given Quantity</a>
%H A000720 A. V. Kumchev, <a href="http://pages.towson.edu/akumchev/M391C.html">
The Distribution of Prime Numbers</a>
%H A000720 J. C. Lagarias, V. S. Miller and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/
~odlyzko/doc/cnt.html">Computing pi(x): The Meissel-Lehmer method</
a>, Math. Comp., 44 (1985), pp. 537-560.
%H A000720 J. C. Lagarias and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/
doc/cnt.html">Computing pi(x): An analytic method</a>, J. Algorithms,
8 (1987), pp. 173-191.
%H A000720 D. J. Lorch, <a href="http://www.bsu.edu/web/math/exchange/03-01/Lorch.pdf">
The Distribution of Primes</a>
%H A000720 B. E. Petersen, <a href="http://oregonstate.edu/~peterseb/misc/docs/pnt.pdf">
Prime Number Theorem(version 1996)</a>
%H A000720 B. E. Petersen, <a href="http://oregonstate.edu/~peterseb/misc/docs/pnt2.pdf">
Prime Number Theorem(version 20020514)</a>
%H A000720 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000720 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000720 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica
de los numeros primos y perfectos</a>
%H A000720 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polprdipi.jpg">
Illustration of initial terms: Divisors and pi(x)</a>
%H A000720 B. Riemann, <a href="http://www.maths.tcd.ie/pub/HistMath/People/Riemann/
Zeta/">On the Number of Prime Numbers</a> 1859, last page (various
transcripts)
%H A000720 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 A000720 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 A000720 A. M. Selvam, <a href="http://lanl.arXiv.org/html/physics/0005067">Quantum-like
Chaos in Prime Number Distribution and in Turbulent Fluid Flows</
a>
%H A000720 A. M. Selvam, <a href="http://redshift.vif.com/JournalFiles/V08NO3PDF/
V08N3SEL.PDF">Quantum-like Chaos in Prime NumberDistribution and
in Turbulent Fluid Flows</a>
%H A000720 J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/bib/pi.bib">
Bibliography on calculation of pi(x)</a>
%H A000720 W. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/ss-a.htm">
The distribution of Prime Numbers</a>
%H A000720 M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/pnt.htm">
the prime number theorem (some references)</a>
%H A000720 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PrimeCountingFunction.html">Link to a section of The World of Mathematics.</
a>
%H A000720 M. Wolf, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/wolfgas.htm">
'Applications of Statistical Mechanics in Prime Number Theory'</a>
%H A000720 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/
PrimePi/03/02">First 50 values of pi(n)</a>
%H A000720 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/
node17.html">Distribution of primes</a>
%H A000720 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000720 The prime number theorem gives the asymptotic expression a(n) ~ n/log(n).
%F A000720 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]
%F A000720 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]
%F A000720 For x >= 55, x/(log(x)+2) < pi(x) < x/(log(x)-4). [Rosser]
%F A000720 For n>1: A138194(n) <= a(n) <= A138195(n) (Tschebyscheff, 1850). - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 04 2008
%F A000720 For n>=3, 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 (benoit7848c(AT)orange.fr), Aug 31 2003
%F A000720 a(n)=A001221(A000142(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jun 03 2005
%F A000720 G.f. sum_{p prime} x^p/(1-x) = b(x)/(1-x), where b(x) is the g.f. for
A010051. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 15
2006
%F A000720 A recursive definition of PrimePi using the LegendrePhi function given
in the Wagon_notes.nb: Pi(n) = Pi(Sqrt(n)) + Phi(n, Pi(Sqrt(n) )
- 1, with Pi(0)=0, Pi(1)=0. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com),
Mar 26 2008
%e A000720 There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.
%p A000720 with(numtheory); A000720 := pi; [ seq(A000720(i),i=1..50) ];
%t A000720 A000720[n_] := PrimePi[n]; Table[ A000720[n], {n, 1, 100} ]
%t A000720 Array[ PrimePi[ # ]&, 100 ]
%o A000720 (PARI) A000720=vector(100,n,omega(n!))
%o A000720 (PARI) vector(300,j,primepi(j)) - Joerg Arndt (arndt(AT)jjj.de), May
09 2008
%o A000720 (Other) sage: [prime_pi(n) for n in xrange(1, 79)] # [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
%Y A000720 Cf. A048989, A006880.
%Y A000720 See also A000040.
%Y A000720 Cf. A132090, A137588.
%Y A000720 Cf. A038107, A099802, A139328.
%Y A000720 Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
[From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
%Y A000720 A143538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 23 2008]
%Y A000720 Cf. A036234. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Mar
23 2009]
%Y A000720 Sequence in context: A071868 A082447 A139789 this_sequence A070549 A074796
A061070
%Y A000720 Adjacent sequences: A000717 A000718 A000719 this_sequence A000721 A000722
A000723
%K A000720 nonn,core,easy,nice
%O A000720 1,3
%A A000720 N. J. A. Sloane (njas(AT)research.att.com).
%E A000720 Additional links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com),
Dec 23 2003
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