A007508 Number of twin prime pairs below 10^n.

2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436

Offset: 1

Author

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

Keywords

• nonn
• nice
• hard
• more

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

Theorem. While g is even, g > 0, number of primes p < x (x is an integer) such that p' = p + g is also prime, could be written as qpg(x) = qcc(x) - (x - pi(x) - pi(x + g) + 1) where qcc(x) is the number of "common composite numbers" c <= x such that c and c' = c + g both are composite (see Example below; I propose it here as a theorem only not to repeat for so-called "cousin"-primes (p; p+4), "sexy"-primes (p; p+6), etc.). - Sergey Pavlov, Apr 08 2021

Examples

			For x = 10, qcc(x) = 4 (since 2 is prime; 4, 6, 8, 10 are even, and no odd 0 < d < 25 such that both d and d' = d + 2 are composite), pi(10) = 4, pi(10 + 2) = 5, but, while v = 2+2 or v = 2-2 would be even, we must add 1; hence, a(1) = qcc(10^1) - (10^1 - pi(10^1) - pi(10^1 + 2) + 1) = 4 - (10 - 4 - 5 + 1) = 2 (trivial). - _Sergey Pavlov_, Apr 08 2021
		

References

• Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 202.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 195.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43-56.
• C. K. Caldwell, An amazing prime heuristic, Table 1.
• 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.
• 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, Physica A 269 (1999) 503-510.
• 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"
• Gareth A. Jones and Alexander K. Zvonkin, Klein's ten planar dessins of degree 11, and beyond, arXiv:2104.12015 [math.GR], 2021. See p. 24.
• 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.
• James Maynard, On the Twin Prime Conjecture, arXiv:1910.14674 [math.NT], 2019, p. 2.
• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
• 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.
• Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only]
• 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, arXiv:1105.2249 [math.NT], 2011; 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, arXiv:math/0605696 [math.NT], 2006.
• 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

Crossrefs

Cf. A001097.

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

Cf. A152051, A347278, A347279.

Programs

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 *)

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

Formula

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

For 1 < n < 19, a(n) ~ e * pi(10^n) / (5*n - 5) = e * A006880(n) / (5*n - 5) where e is Napier's constant, see A001113 (probably, so is for any n > 18; we use n > 1 to avoid division by zero). - Sergey Pavlov, Apr 07 2021

For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(10^n + 2) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that c and c' = c + 2 both are composite (trivial). - Sergey Pavlov, Apr 08 2021

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