A070076 -id:A070076 - cp's OEIS frontend

A001097 Twin primes.

3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643

Offset: 1

N. J. A. Sloane, Mar 15 1996

Union of A001359 and A006512.
The only twin primes that are Fibonacci numbers are 3, 5 and 13 [MacKinnon]. - Emeric Deutsch, Apr 24 2005
(p, p+2) are twin primes if and only if p + 2 can be represented as the sum of two primes. Brun (1919): Even if there are infinitely many twin primes, the series of all twin prime reciprocals does converges to [Brun's constant] (A065421). Clement (1949): For every n > 1, (n, n+2) are twin primes if and only if 4((n-1)! + 1) == -n (mod n(n+2)). - Stefan Steinerberger, Dec 04 2005
A164292(a(n)) = 1. - Reinhard Zumkeller, Mar 29 2010
The 100355-digit numbers 65516468355 * 2^333333 +- 1 are currently the largest known twin prime pair. They were discovered by Twin Prime Search and Primegrid in August 2009. - Paul Muljadi, Mar 07 2011
For every n > 2, the pair (n, n+2) is a twin prime if and only if ((n-1)!!)^4 == 1 (mod n*(n+2)). - Thomas Ordowski, Aug 15 2016
The term "twin primes" ("primzahlzwillinge", in German) was coined by the German mathematician Paul Gustav Samuel Stäckel (1862-1919) in 1916. Brun (1919) used the same term in French ("nombres premiers jumeaux"). Glaisher (1878) and Hardy and Littlewood (1923) used the term "prime-pairs". The term "twin primes" in English was used by Dantzig (1930). - Amiram Eldar, May 20 2023

Tobias Dantzig, Number: The Language of Science, Macmillan, 1930.
Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, pp. 259-265.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 132.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
Viggo Brun, La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie, Bull Sci. Math. 43 (1919), 100-104 and 124-128.
J. P. Delahaye, Premiers jumeaux: frères ennemis? [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
J. C. Evard, Twin primes and their applications. [Cached copy on the Wayback Machine]
J. C. Evard, Twin primes and their applications. [Local cached copy]
J. C. Evard, Twin primes and their applications. [Pdf file of cached copy]
J. W. L. Glaisher, An enumeration of prime-pairs, Messenger of Mathematics, Vol. 8 (1878), pp. 28-33.
Andrew Granville, Primes in intervals of bounded length, Joint Math Meeting, Current Events Bulletin, Baltimore, Friday, Jan 17 2014.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, No. 1 (1923), pp. 1-70; alternative link.
Nick MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78.
James Maynard, Small gaps between primes, Annals of Mathematics, Second series, Vol. 181, No. 1 (2015), pp. 383-413; arXiv preprint, arXiv:1311.4600 [math.NT], 2013-2019.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes, arXiv:1402.0811 [math.NT], 2014.
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, arXiv:1407.4897 [math.NT], 2014.
Paul Stäckel, Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Klasse (in German), Abt. A, Bd. 10 (1916), pp. 1-47. See p. 22.
Eric Weisstein's World of Mathematics, Twin Primes.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Volume 179, Issue 3 (2014), Pages 1121-1174.
Index entries for "core" sequences.
Index entries for primes, gaps between.

Cf. A070076, A001359, A006512, A164292. See A077800 for another version.

a001097 n = a001097_list !! (n-1)
a001097_list = filter ((== 1) . a164292) [1..]
-- Reinhard Zumkeller, Feb 03 2014, Nov 27 2011

A001097 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isprime(a) and ( isprime(a-2) or isprime(a+2) ) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Feb 19 2015

Select[ Prime[ Range[120]], PrimeQ[ # - 2] || PrimeQ[ # + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Union[Flatten[Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] == 2&]]] (* Harvey P. Dale, Aug 19 2015 *)

isA001097(n) = (isprime(n) && (isprime(n+2) || isprime(n-2))) \\ Michael B. Porter, Oct 29 2009

a(n)=if(n==1,return(3));my(p);forprime(q=3,default(primelimit), if(q-p==2 && (n-=2)<0, return(if(n==-1,q,p)));p=q) \\ Charles R Greathouse IV, Aug 22 2012

list(lim)=my(v=List([3]),p=5); forprime(q=7,lim, if(q-p==2, listput(v,p); listput(v,q)); p=q); if(p+2>lim && isprime(p+2), listput(v,p)); Vec(v) \\ Charles R Greathouse IV, Mar 17 2017

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    yield 3
    p, q = 5, 7
    while True:
        if q - p == 2: yield from [p, q]
        p, q = q, nextprime(q)
print(list(islice(agen(), 58))) # Michael S. Branicky, Apr 30 2022

A077800 List of twin primes {p, p+2}.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619

Offset: 1

N. J. A. Sloane, Dec 03 2002

Union (with repetition) of A001359 and A006512.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
Jean-Paul Delahaye, Premiers jumeaux: frères ennemis? [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.
Jean-Claude Evard, Twin primes and their applications. [Cached copy on the Wayback Machine]
Jean-Claude Evard, Twin primes and their applications. [Local cached copy]
Jean-Claude Evard, Twin primes and their applications. [Pdf file of cached copy]
Dave Platt and Tim Trudgian, Improved bounds on Brun's constant, in: David H. Bailey et al. (eds), From Analysis to Visualization, JBCC 2017, Springer Proceedings in Mathematics & Statistics, Vol 313, Springer, Cham, 2020, preprint, arXiv:1803.01925 [math.NT], 2018.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 114.
Hayden Tronnolone, A tale of two primes, COLAUMS Space, #3, 2013.
Wikipedia, Twin prime.
Index entries for primes, gaps between

Cf. A065421, A070076, A095958. See A001097 for another version.

a077800 n = a077800_list !! (n-1)
a077800_list = concat $ zipWith (\p q -> if p == q+2 then [q,p] else [])
                                (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Nov 27 2011

Sort[ Join[ Select[ Prime[ Range[ 115]], PrimeQ[ # - 2] &], Select[ Prime[ Range[ 115]], PrimeQ[ # + 2] &]]] (* Robert G. Wilson v, Jun 09 2005 *)
Select[ Partition[ Prime@ Range@ 115, 2, 1], #[[1]] + 2 == #[[2]] &] // Flatten
Flatten[Select[{#, # + 2} & /@Prime[Range[1000]], PrimeQ[Last[#]]&]] (* Vincenzo Librandi, Nov 01 2012 *)
Splice[{#,#+2}]& /@ Select[Prime[Range[PrimePi[619]]], PrimeQ[#+2]&] (* Oliver Seipel, Sep 04 2024 *)

p=2;forprime(q=3,1e3,if(q-p==2,print1(p", "q", "));p=q) \\ Charles R Greathouse IV, Mar 22 2013

Sum_{n>=1} 1/a(n) is in the interval (1.840503, 2.288490) (Platt and Trudgian, 2020). The conjectured value based on assumptions about the distribution of twin primes is A065421. - Amiram Eldar, Oct 15 2020

A007508 Number of twin prime pairs below 10^n.

Original entry on oeis.org

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

Offset: 1

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

"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

			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

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).

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

Cf. A001097.
Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).
Cf. A152051, A347278, A347279.

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 *)

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

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

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

A099609 Naive list of twin primes (A077800 prefixed by 2, 3).

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641

Offset: 1

N. J. A. Sloane, Nov 18 2004

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. P. Delahaye, Premiers jumeaux: freres ennemis? [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.
J. C. Evard, Twin primes and their applications [Cached copy on the Wayback Machine]
J. C. Evard, Twin primes and their applications [Local cached copy]
J. C. Evard, Twin primes and their applications [Pdf file of cached copy]
Wikipedia, Twin prime

Cf. A070076, A077800. See A001097 for another version.

Select[Partition[#, 2, 1] &@ Prime@ Range@ 120, First@ Differences@ # <= 2 &] // Flatten (* Michael De Vlieger, Mar 18 2017 *)

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A001097 Twin primes.

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A077800 List of twin primes {p, p+2}.

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A007508 Number of twin prime pairs below 10^n.

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A099609 Naive list of twin primes (A077800 prefixed by 2, 3).

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A120120 Number of n-digit prime quadruplets.

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