%I A007508 M1855
%S A007508 2,8,35,205,1224,8169,58980,440312,3424506,27412679,224376048,
%T A007508 1870585220,15834664872,135780321665,1177209242304,10304185697298,
%U A007508 90948839353159,808675888577436
%N A007508 Number of twin primes < 10^n.
%C A007508 "At the present time (2001), Thomas Nicely has reached pi_2(3*10^15) 
               and his value is confirmed by Pascal Sebah who made a new computation 
               from scratch and up to pi_2(5*10^15) [ = 5357875276068] with an independent 
               implementation."
%C A007508 Though the first paper contributed by D. A. Goldston was reported to 
               be flawed, the more recent one (with other coauthors) maintains and 
               substantiates the result. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Aug 19 2005
%D A007508 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007508 T. R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, 
               Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
%D A007508 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 202.
%H A007508 R. F. Arenstorf, <a href="http://arXiv.org/abs/math.NT/0405509">There 
               Are Infinitely Many Prime Twins</a>
%H A007508 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub024.html">
               Irregularities in the distribution of primes and twin primes</a>
%H A007508 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php/TwinPrimeConjecture.html">Twin prime conjecture</a>
%H A007508 T. H. Chan, <a href="http://arXiv.org/abs/math.NT/0503441">A note on 
               Primes in Short Intervals</a>
%H A007508 J. Derbyshire, <a href="http://olimu.com/Notes/GoldstonYildirim.htm">
               Goldston & Yildirim's Result</a>
%H A007508 P. Erdos, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/
               1028997963">Some Unsolved Problems</a>
%H A007508 G. H. Gadiyar & R. Padma, <a href="http://arXiv.org/abs/hep-th/9806061">
               Renormalisation and the density of prime pairs</a>
%H A007508 G. H. Gadiyar & R. Padma, <a href="http://www.maths.ex.ac.uk/~mwatkins/
               zeta/padma.pdf">Ramanujan-Fourier series, the Wiener-Khintchine formula 
               and the distribution of prime pairs</a>
%H A007508 D. A. Goldston, J. Pintz & C. Y. Yildirim, <a href="http://arXiv.org/
               math.NT/0508185">Primes in Tuples, I</a>
%H A007508 D. A. Goldston, J. Pintz & C. Y. Yildirim, <a href="http://www.math.sjsu.edu/
               ~goldston/gpy02-08-05.pdf">Small Gaps Between Primes, II</a>
%H A007508 D. A. Goldston, J. Pintz & C. Y. Yildirim, <a href="http://arXiv.org/
               abs/math/0512436">The Path to Recent Progress on Small Gaps Between 
               Primes</a>
%H A007508 D. A. Goldston & C. Y. Yildirim, <a href="http://arXiv.org/math.NT/0504336">
               Small Gaps Between Primes, I</a>
%H A007508 D. A. Goldston & C. Yildirim, <a href="http://aimath.org/primegaps/goldston_tech">
               Small gaps between consecutive primes</a>
%H A007508 D. A. Goldston et al., <a href="http://arXiv.org/abs/math.NT/0506067">
               Small gaps between primes or almost primes</a>
%H A007508 D. A. Goldston et al., <a href="http://arXiv.org/abs/math.NT/0505300">
               Small Gaps between Primes Exist</a>
%H A007508 Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/
               Constants/Primes/twin.html">Introduction to Twin Primes and Brun's 
               Constant</a>
%H A007508 A. Granville & K. Soundararajan, <a href="http://aimath.org/primegaps/
               residueerror">On the error in Goldston and Yildirim's "Small gaps 
               between consecutive primes"</a>
%H A007508 P. F. Kelly & F. Pilling, <a href="http://arXiv.org/pdf/math.NT/0103191">
               Characterization of the Distribution of Twin Primes</a>
%H A007508 P. F. Kelly & T. Pilling, <a href="http://arXiv.org/abs/math.NT/0104205">
               Implications of a New Characterization of the Distribution of Twin 
               Primes</a>
%H A007508 P. F. Kelly & T. Pilling, <a href="http://arXiv.org/abs/math.NT/0106223">
               Discrete Reanalysis of a New Model of the Distribution of Twin Primes</
               a>
%H A007508 Thomas R. Nicely, <a href="http://www.trnicely.net/">Home page</a>. Has 
               extensive tables.
%H A007508 Nova Science, <a href="http://www.pbs.org/wgbh/nova/sciencenow/3302/02.html">
               Twin Prime Conjecture</a>
%H A007508 Tomas Oliveira e Silva, <a href="http://www.ieeta.pt/~tos/primes.html">
               Tables of values of pi(x) and of pi2(x)</a> [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               Dec 18 2008]
%H A007508 J. Richstein, <a href="http://www.informatik.uni-giessen.de/staff/richstein/
               res/tp-en.html">Computing the number of twin primes up to 10^14</
               a>
%H A007508 J. Richstein, <a href="http://www.mscs.dal.ca/~joerg/res/tp-en.html">
               Computing the number of twin primes up to 10^14</a>
%H A007508 K. Soundararajan, <a href="http://arXiv.org/abs/math.NT/0606408">The 
               distribution of prime numbers</a>
%H A007508 K. Soundararajan, <a href="http://arXiv.org/abs/math.NT/0605696">Small 
               gaps between prime numbers:The work of Goldston-Pintz-Yildirim</a>
%H A007508 K. Soundararajan, <a href="http://www.ams.org/bull/2007-44-01/S0273-0979-06-01142-6/
               home.html">Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim</
               a>
%H A007508 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TwinPrimes.html">Link to a section of The World of Mathematics.</
               a>
%H A007508 Eric Weisstein, Mathworld Headline News, <a href="http://mathworld.wolfram.com/
               news/2004-06-09/twinprimes">Twin Primes Proof Proffered</a>
%H A007508 M. Wolf, <a href="http://arXiv.org/abs/math/0105211">Some Remarks on 
               the Distribution of twin Primes</a>
%H A007508 C. Yildirim & D. Goldston, <a href="http://www.math.boun.edu.tr/instructors/
               yildirim/yildirimtech.htm">Small gaps between consecutive primes</
               a>
%H A007508 <a href="Sindx_Pri.html#primepop">Index entries for sequences related 
               to numbers of primes in various ranges</a>
%F A007508 Partial sums of A070076(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 11 2004
%Y A007508 Cf. A001097.
%Y A007508 Sequence in context: A030820 A030972 A020009 this_sequence A122674 A076122 
               A123290
%Y A007508 Adjacent sequences: A007505 A007506 A007507 this_sequence A007509 A007510 
               A007511
%K A007508 nonn,nice,hard
%O A007508 1,1
%A A007508 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A007508 pi2(10^15) due to Nicely and Szymanski, contributed by Eric Weisstein 
               (eric(AT)weisstein.com)
%E A007508 pi2(10^16) due to Pascal Sebah, contributed by Robert G. Wilson v (rgwv(AT)rgwv.com), 
               Aug 22 2002
%E A007508 Added a(17)-a(18) computed by Tomas Oliveira e Silva and link to his 
               web site. M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 18 2008