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

%I M1018

%S 0,1,2,4,6,11,18,31,54,97,172,309,564,1028,1900,3512,6542,12251,23000,

%T 43390,82025,155611,295947,564163,1077871,2063689,3957809,7603553,

%U 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

%N Number of primes <= 2^n.

%C 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

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

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

%H Charles R Greathouse IV and Douglas B. Staple, <a href="/A007053/b007053.txt">Table of n, a(n) for n = 0..86</a> [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 RH, verified unconditionally by Douglas B. Staple, and a(81)-a(86) from Douglas B. Staple]

%H Andrew R. Booker, <a href="http://primes.utm.edu/nthprime/">The Nth Prime Page</a>

%H S. W. Golomb, <a href="/A007053/a007053.pdf">Letter to N. J. A. Sloane, Jul. 1991</a>

%H Thomas R. Nicely, <a href="http://www.trnicely.net/index.html">Some Results of Computational Research in Prime Numbers</a>

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/primes.html">Tables of values of pi(x) and of pi2(x)</a>

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/bib/5.4.pdf">Computing pi(x): the combinatorial method</a>, Revista Do Detua, Vol. 4, No 6, March 2006.

%H Douglas B. Staple, <a href="http://arxiv.org/abs/1503.01839">The combinatorial algorithm for computing pi(x)</a>, arXiv:1503.01839 [math.NT], 2015.

%H <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>

%F a(n) = A060967(2n). - _R. J. Mathar_, Sep 15 2012

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

%t Table[PrimePi[2^n], {n, 0, 46}] (* _Robert G. Wilson v_ *)

%o (PARI) a(n) = primepi(1<<n); \\ _John W. Nicholson_, May 16 2011

%Y Cf. A006880, A036378.

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_, S. W. Golomb

%E More terms from _Jud McCranie_

%E 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.

%E Extended to n = 86 by _Douglas B. Staple_, Dec 18 2014

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Last modified May 24 20:53 EDT 2019. Contains 323534 sequences. (Running on oeis4.)