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A007508 Number of twin prime pairs below 10^n.
(Formerly M1855)
46
2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

"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."

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, Aug 19 2005

REFERENCES

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 202.

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

LINKS

Table of n, a(n) for n=1..18.

R. P. Brent, Irregularities in the distribution of primes and twin primes

C. K. Caldwell, The Prime Glossary, Twin prime conjecture

T. H. Chan, A note on Primes in Short Intervals, arXiv:math/0503441 [math.NT], 2005.

J. Derbyshire, Goldston & Yildirim's Result

P. Erdős, Some Unsolved Problems, Michigan Math. J., Volume 4, Issue 3 (1957), 291-300.

G. H. Gadiyar & R. Padma, Renormalisation and the density of prime pairs, arXiv:hep-th/9806061, 1998.

G. H. Gadiyar & R. Padma, Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs

D. A. Goldston, J. Pintz & C. Y. Yildirim, Primes in Tuples, I, arXiv:math/0508185 [math.NT], 2005.

D. A. Goldston, J. Pintz & C. Y. Yildirim, Small Gaps Between Primes, II

D. A. Goldston, J. Pintz & C. Y. Yildirim, The Path to Recent Progress on Small Gaps Between Primes, arXiv:math/0512436 [math.NT], 2005-2006.

D. A. Goldston & C. Y. Yildirim, Small Gaps Between Primes, I, arXiv:math/0504336 [math.NT], 2005.

D. A. Goldston & C. Yildirim, Small gaps between consecutive primes

D. A. Goldston et al., Small gaps between primes or almost primes, arXiv:math/0506067 [math.NT], 2005.

D. A. Goldston et al., Small Gaps between Primes Exist, arXiv:math/0505300 [math.NT], 2005.

Xavier Gourdon and Pascal Sebah, Introduction to Twin Primes and Brun's Constant

A. Granville & K. Soundararajan, On the error in Goldston and Yildirim's "Small gaps between consecutive primes"

P. F. Kelly & F. Pilling, Characterization of the Distribution of Twin Primes, arXiv:math/0103191 [math.NT], 2001.

P. F. Kelly & T. Pilling, Implications of a New Characterization of the Distribution of Twin Primes, arXiv:math/0104205 [math.NT], 2001.

P. F. Kelly & T. Pilling, Discrete Reanalysis of a New Model of the Distribution of Twin Primes, arXiv:math/0106223 [math.NT], 2001.

Thomas R. Nicely, Home page. Has extensive tables.

Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.

Nova Science, Twin Prime Conjecture

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From M. F. Hasler, Dec 18 2008]

J. Richstein, Computing the number of twin primes up to 10^14

J. Richstein, Computing the number of twin primes up to 10^14

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.

K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], 2006.

K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim

K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), 1-18.

Eric Weisstein's World of Mathematics, Twin Primes

Eric Weisstein, Mathworld Headline News, Twin Primes Proof Proffered

M. Wolf, Some Remarks on the Distribution of twin Primes, arXiv:math/0105211 [math.NT], 2001.

C. Yildirim & D. Goldston, Small gaps between consecutive primes

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

FORMULA

Partial sums of A070076(n). - Lekraj Beedassy, Jun 11 2004

MATHEMATICA

ile = 2; Do[Do[If[(PrimeQ[2 n - 1]) && (PrimeQ[2 n + 1]), ile = ile + 1], {n, 5*10^m, 5*10^(m + 1)}]; Print[{m, ile}], {m, 0, 7}] (* Artur Jasinski, Oct 24 2011 *)

PROG

(PARI) a(n)=my(s, p=2); forprime(q=3, 10^n, if(q-p==2, s++); p=q); s \\ Charles R Greathouse IV, Mar 21 2013

CROSSREFS

Cf. A001097.

Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).

Sequence in context: A213227 A020009 A257555 * A122674 A203762 A185635

Adjacent sequences:  A007505 A007506 A007507 * A007509 A007510 A007511

KEYWORD

nonn,nice,hard

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

pi2(10^15) due to Nicely and Szymanski, contributed by Eric W. Weisstein

pi2(10^16) due to Pascal Sebah, contributed by Robert G. Wilson v, Aug 22 2002

Added a(17)-a(18) computed by Tomás Oliveira e Silva and link to his web site. - M. F. Hasler, Dec 18 2008

Definition corrected by Max Alekseyev, Oct 25 2010

a(16) corrected by Dana Jacobsen, Mar 28 2014

STATUS

approved

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Last modified January 17 17:17 EST 2019. Contains 319235 sequences. (Running on oeis4.)