Search: id:A000720 Results 1-1 of 1 results found. %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, Table of n, pi(n) for n = 1..100000 %H A000720 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]. %H A000720 S. Bennett, The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem %H A000720 C. Bonanno & M. S. Mega, Toward a dynamic model for prime numbers %H A000720 D. M. Bressoud, Review of "The Prime Number Theorem" by G. J. O. Jameson %H A000720 B. Brubaker, The Prime Number Theorem %H A000720 C. K. Caldwell, The Prime Glossary, Prime number theorem %H A000720 C. K. Caldwell, How Many Primes Are There %H A000720 W. W. L. Chen, Distribution of Prime Numbers %H A000720 M. Deleglise, Computation of large values of pi(x) %H A000720 Encyclopedia Britannica, The Prime Number Theorem %H A000720 G. H. Hardy & J. E. Littlewood, Contributions To The Theory Of The Riemann Zeta-Function And The Theory Of The Distribution Of Primes %H A000720 M. Hassani, Approximation of pi(x) by Psi(x), J. Inequ. Pure Appl. Math. 7 (2006) vol. 1, #7 %H A000720 Y.-C. Kim, Note on the Prime Number Theorem %H A000720 T. V. Kolev, On the number of Prime Numbers less than a Given Quantity %H A000720 A. V. Kumchev, The Distribution of Prime Numbers %H A000720 J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560. %H A000720 J. C. Lagarias and A. M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191. %H A000720 D. J. Lorch, The Distribution of Primes %H A000720 B. E. Petersen, Prime Number Theorem(version 1996) %H A000720 B. E. Petersen, Prime Number Theorem(version 20020514) %H A000720 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. %H A000720 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000720 O. E. Pol, Determinacion geometrica de los numeros primos y perfectos %H A000720 O. E. Pol, Illustration of initial terms: Divisors and pi(x) %H A000720 B. Riemann, On the Number of Prime Numbers 1859, last page (various transcripts) %H A000720 J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy) %H A000720 S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime %H A000720 A. M. Selvam, Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows %H A000720 A. M. Selvam, Quantum-like Chaos in Prime NumberDistribution and in Turbulent Fluid Flows %H A000720 J. O. Shallit, Bibliography on calculation of pi(x) %H A000720 W. R. Watkins, The distribution of Prime Numbers %H A000720 M. R. Watkins, the prime number theorem (some references) %H A000720 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000720 M. Wolf, 'Applications of Statistical Mechanics in Prime Number Theory' %H A000720 Wolfram Research, First 50 values of pi(n) %H A000720 D. J. Wright, Distribution of primes %H A000720 Index entries for "core" sequences %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 Search completed in 0.005 seconds