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 A001097 Twin primes. 243

%I

%S 3,5,7,11,13,17,19,29,31,41,43,59,61,71,73,101,103,107,109,137,139,

%T 149,151,179,181,191,193,197,199,227,229,239,241,269,271,281,283,311,

%U 313,347,349,419,421,431,433,461,463,521,523,569,571,599,601,617,619,641,643

%N Twin primes.

%C Union of A001359 and A006512.

%C The only twin primes that are Fibonacci numbers are 3, 5 and 13 [MacKinnon]. - _Emeric Deutsch_, Apr 24 2005

%C (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

%C A164292(a(n)) = 1. - _Reinhard Zumkeller_, Mar 29 2010

%C 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

%C 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

%D P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, pp. 259-265.

%H T. D. Noe, <a href="/A001097/b001097.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.

%H J. P. Delahaye, <a href="http://www.lifl.fr/~delahaye/SIME/JPD/PLS_Nb_premiers_jumeaux.pdf">Premiers jumeaux: frères ennemis?</a> [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.

%H Harvey Dubner, <a href="http://www.emis.de/journals/JIS/VOL8/Dubner/dubner71.html">Twin Prime Statistics</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.

%H J. C. Evard, <a href="http://web.archive.org/web/20110726012847/http://www.math.utoledo.edu/~jevard/Page012.htm">Twin primes and their applications</a> [Cached copy on the Wayback Machine]

%H J. C. Evard, <a href="/A077800/a077800.html">Twin primes and their applications</a> [Local cached copy]

%H J. C. Evard, <a href="/A077800/a077800.pdf">Twin primes and their applications</a> [Pdf file of cached copy]

%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/BaltimoreLecture.pdf">Primes in intervals of bounded length</a>, Joint Math Meeting, Current Events Bulletin, Baltimore, Friday, Jan 17 2014.

%H N. MacKinnon, <a href="http://www.jstor.org/stable/2695779">Problem 10844</a>, Amer. Math. Monthly 109, (2002), p. 78.

%H James Maynard, <a href="http://arxiv.org/abs/1311.4600">Small gaps between primes</a>, arXiv:1311.4600 [math.NT], 2013, Annals of Mathematics, to appear.

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

%H D. H. J. Polymath, <a href="http://arxiv.org/abs/1402.0811">New equidistribution estimates of Zhang type, and bounded gaps between primes</a>, arXiv:1402.0811 [math.NT], 2014.

%H D. H. J. Polymath, <a href="http://arxiv.org/abs/1407.4897">Variants of the Selberg sieve, and bounded intervals containing many primes</a>, arXiv:1407.4897 [math.NT], 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>

%H Yitang Zhang, <a href="http://dx.doi.org/10.4007/annals.2014.179.3.7">Bounded gaps between primes</a>, Annals of Mathematics, Pages 1121-1174, Volume 179 (2014), Issue 3.

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%p A001097 := proc(n)

%p option remember;

%p if n = 1 then

%p 3;

%p else

%p for a from procname(n-1)+1 do

%p if isprime(a) and ( isprime(a-2) or isprime(a+2) ) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Feb 19 2015

%t Select[ Prime[ Range], PrimeQ[ # - 2] || PrimeQ[ # + 2] &] (* _Robert G. Wilson v_, Jun 09 2005 *)

%t Union[Flatten[Select[Partition[Prime[Range],2,1],#[]-#[] == 2&]]] (* _Harvey P. Dale_, Aug 19 2015 *)

%o (PARI) isA001097(n) = (isprime(n) && (isprime(n+2) || isprime(n-2))) \\ _Michael B. Porter_, Oct 29 2009

%o (PARI) 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

%o (PARI) list(lim)=my(v=List(),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

%o (Haskell)

%o a001097 n = a001097_list !! (n-1)

%o a001097_list = filter ((== 1) . a164292) [1..]

%o -- _Reinhard Zumkeller_, Feb 03 2014, Nov 27 2011

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

%K nonn,core,nice

%O 1,1

%A _N. J. A. Sloane_, Mar 15 1996

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