1,3

Partial sums of A010051 (characteristic function of primes). - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 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 (jsondow(AT)alumni.princeton.edu), Dec 27 2004

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.

See the additional references and links mentioned in A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Equals row sums of triangle A143538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 23 2008]

a(n) = A036234(n)-1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Mar 23 2009]

((10^n)^2)/(ln((10^n)!)) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Jul 15 2009]

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.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.

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.

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

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 5.

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.

R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.

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

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.1. (For inequalities, etc.)

W. Narkiewicz, The Development of Prime Number Theory, Springer-Verlag 2000.

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

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.

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).

G. Tenebaum and M. Mendes France, Prime Numbers and Their Distribution, AMS Providence RI 1999

Wikipedia, Prime Number Theorem.

Daniel Forgues, Table of n, pi(n) for n = 1..100000

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].

S. Bennett, The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem

C. Bonanno & M. S. Mega, Toward a dynamic model for prime numbers

D. M. Bressoud, Review of "The Prime Number Theorem" by G. J. O. Jameson

B. Brubaker, The Prime Number Theorem

C. K. Caldwell, The Prime Glossary, Prime number theorem

C. K. Caldwell, How Many Primes Are There

W. W. L. Chen, Distribution of Prime Numbers

M. Deleglise, Computation of large values of pi(x)

Encyclopedia Britannica, The Prime Number Theorem

G. H. Hardy & J. E. Littlewood, Contributions To The Theory Of The Riemann Zeta-Function And The Theory Of The Distribution Of Primes

M. Hassani, Approximation of pi(x) by Psi(x), J. Inequ. Pure Appl. Math. 7 (2006) vol. 1, #7

Y.-C. Kim, Note on the Prime Number Theorem

T. V. Kolev, On the number of Prime Numbers less than a Given Quantity

A. V. Kumchev, The Distribution of Prime Numbers

J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.

J. C. Lagarias and A. M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191.

D. J. Lorch, The Distribution of Primes

B. E. Petersen, Prime Number Theorem(version 1996)

B. E. Petersen, Prime Number Theorem(version 20020514)

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos

O. E. Pol, Illustration of initial terms: Divisors and pi(x)

B. Riemann, On the Number of Prime Numbers 1859, last page (various transcripts)

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

A. M. Selvam, Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows

A. M. Selvam, Quantum-like Chaos in Prime NumberDistribution and in Turbulent Fluid Flows

J. O. Shallit, Bibliography on calculation of pi(x)

W. R. Watkins, The distribution of Prime Numbers

M. R. Watkins, the prime number theorem (some references)

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

M. Wolf, 'Applications of Statistical Mechanics in Prime Number Theory'

Wolfram Research, First 50 values of pi(n)

D. J. Wright, Distribution of primes

Index entries for "core" sequences

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 (reinhard.zumkeller(AT)gmail.com), Mar 04 2008

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

a(n)=A001221(A000142(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 15 2006

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

There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.

with(numtheory); A000720 := pi; [ seq(A000720(i), i=1..50) ];

A000720[n_] := PrimePi[n]; Table[ A000720[n], {n, 1, 100} ]

Array[ PrimePi[ # ]&, 100 ]

(PARI) A000720=vector(100, n, omega(n!))

(PARI) vector(300, j, primepi(j)) - Joerg Arndt (arndt(AT)jjj.de), May 09 2008

(Other) sage: [prime_pi(n) for n in xrange(1, 79)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]

Cf. A048989, A006880.

See also A000040.

Cf. A132090, A137588.

Cf. A038107, A099802, A139328.

Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

A143538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 23 2008]

Cf. A036234. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Mar 23 2009]

Sequence in context: A071868 A082447 A139789 this_sequence A070549 A074796 A061070

Adjacent sequences: A000717 A000718 A000719 this_sequence A000721 A000722 A000723

nonn,core,easy,nice

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

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