A007053 Number of primes <= 2^n. (Formerly M1018)

0, 1, 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, 155611, 295947, 564163, 1077871, 2063689, 3957809, 7603553, 14630843, 28192750, 54400028, 105097565, 203280221, 393615806, 762939111, 1480206279, 2874398515, 5586502348, 10866266172, 21151907950, 41203088796, 80316571436, 156661034233, 305761713237, 597116381732, 1166746786182, 2280998753949, 4461632979717, 8731188863470, 17094432576778, 33483379603407, 65612899915304, 128625503610475

Offset

0,3

Comments

Conjecture: The number 4 is the only perfect power in this sequence. In other words, it is impossible to have a(n) = x^m for some integers n > 3, m > 1 and x > 1. - Zhi-Wei Sun, Sep 30 2015

References

Jens Franke et al., pi(10^24), Posting to the Number Theory Mailing List, Jul 29 2010.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

David Baugh, Table of n, a(n) for n = 0..92 (terms n = 87..92 found using Kim Walisch's primecount program, terms n = 0..86 from Charles R Greathouse IV and Douglas B. Staple, [a(0)-a(75) from Tomás Oliveira e Silva, a(76)-a(77) from Jens Franke et al., Jul 29 2010, a(78)-a(80) from Jens Franke et al. on the Riemann Hypothesis, verified unconditionally by Douglas B. Staple, and a(81)-a(86) from Douglas B. Staple])

Andrew R. Booker, The Nth Prime Page

S. W. Golomb, Letter to N. J. A. Sloane, Jul. 1991

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [Local copy, pdf only]

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista Do Detua, Vol. 4, No 6, March 2006.

Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015.

Index entries for sequences related to numbers of primes in various ranges

Formula

a(n) = A060967(2n). - R. J. Mathar, Sep 15 2012

Example

pi(2^3)=4 since first 4 primes are 2,3,5,7 all <= 2^3 = 8.

Mathematica

Table[PrimePi[2^n], {n, 0, 46}] (* Robert G. Wilson v *)

Prog

(PARI) a(n) = primepi(1<<n); \\ John W. Nicholson, May 16 2011

Crossrefs

Cf. A006880, A036378.

Sequence in context: A168445 A328669 A185192 * A005684 A260697 A018167

Adjacent sequences: A007050 A007051 A007052 * A007054 A007055 A007056

Keyword

nonn,nice

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, S. W. Golomb

Extensions

More terms from Jud McCranie

Extended to n = 52 by Warren D. Smith, Dec 11 2000, computed with Meissel-Lehmer-Legendre inclusion exclusion formula code he wrote back in 1985, recently re-run.

Extended to n = 86 by Douglas B. Staple, Dec 18 2014

Status

approved