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A007508
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Number of twin prime pairs below 10^n.
(Formerly M1855)
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33
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2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304185697298, 90948839353159, 808675888577436
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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"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
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REFERENCES
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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.
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).
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LINKS
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Table of n, a(n) for n=1..18.
R. F. Arenstorf, There Are Infinitely Many Prime Twins
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
J. Derbyshire, Goldston & Yildirim's Result
P. Erdos, Some Unsolved Problems
G. H. Gadiyar & R. Padma, Renormalisation and the density of prime pairs
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
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
D. A. Goldston & C. Y. Yildirim, Small Gaps Between Primes, I
D. A. Goldston & C. Yildirim, Small gaps between consecutive primes
D. A. Goldston et al., Small gaps between primes or almost primes
D. A. Goldston et al., Small Gaps between Primes Exist
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
P. F. Kelly & T. Pilling, Implications of a New Characterization of the Distribution of Twin Primes
P. F. Kelly & T. Pilling, Discrete Reanalysis of a New Model of the Distribution of Twin Primes
Thomas R. Nicely, Home page. Has extensive tables.
Nova Science, Twin Prime Conjecture
Tomas 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
K. Soundararajan, The distribution of prime numbers
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
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
C. Yildirim & D. Goldston, Small gaps between consecutive primes
Index entries for sequences related to numbers of primes in various ranges
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FORMULA
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Partial sums of A070076(n). - Lekraj Beedassy, Jun 11 2004
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001097.
Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).
Sequence in context: A030972 A213227 A020009 * A122674 A203762 A185635
Adjacent sequences: A007505 A007506 A007507 * A007509 A007510 A007511
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KEYWORD
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nonn,nice,hard
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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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 Tomas Oliveira e Silva and link to his web site. M. F. Hasler, Dec 18 2008
Definition corrected by Max Alekseyev, Oct 25 2010
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STATUS
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approved
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