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%I M3763 #161 Feb 05 2024 00:59:01
%S 5,7,13,19,31,43,61,73,103,109,139,151,181,193,199,229,241,271,283,
%T 313,349,421,433,463,523,571,601,619,643,661,811,823,829,859,883,1021,
%U 1033,1051,1063,1093,1153,1231,1279,1291,1303,1321,1429,1453,1483,1489,1609
%N Greater of twin primes.
%C Also primes that are the sum of two primes (which is possible only if 2 is one of the primes). - _Cino Hilliard_, Jul 02 2004, edited by _M. F. Hasler_, Nov 14 2019
%C The set of greater of twin primes larger than five is a proper subset of the set of primes of the form 3n + 1 (A002476). - _Paul Muljadi_, Jun 05 2008
%C Smallest prime > n-th isolated composite. - _Juri-Stepan Gerasimov_, Nov 07 2009
%C Subsequence of A175075. Union of a(n) and sequence A175080 is A175075. - _Jaroslav Krizek_, Jan 30 2010
%C A164292(a(n))=1; A010051(a(n)+2)=0 for n > 1. - _Reinhard Zumkeller_, Mar 29 2010
%C Omega(n) = Omega(n-2); d(n) = d(n-2). - _Juri-Stepan Gerasimov_, Sep 19 2010
%C Aside from the first term, all subsequent terms have digital root 1, 4, or 7. - _J. W. Helkenberg_, Jul 24 2013
%C Also primes p with property that the sum of the successive gaps between primes <= p is a prime number. - _Robert G. Wilson v_, Dec 19 2014
%C The phrase "x is an element of the {primes, positive integers} and there {exist no, exist} elements a,b of {1 and primes, primes}: a+b=x" determines A133410, A067829, A025584, A006512, A166081, A014092, A014091 and A038609 for the first few hundred terms with only de-duplication or omitting/including 3, 4 and 6 in the case of A166081/A014091 and one case of omitting/including 3 given 1 isn't prime. - _Harry G. Coin_, Nov 25 2015
%C The yet unproved Twin Prime Conjecture states that this sequence is infinite. - _M. F. Hasler_, Nov 14 2019
%D See A001359 for further references and links.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006512/b006512.txt">Table of n, a(n) for n = 1..10000</a>
%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 R. K. Guy, <a href="/A006511/a006511.pdf">Letter to N. J. A. Sloane, Jun 1991</a>
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Twin_prime#History">Twin prime</a>.
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%p for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i+1)}); fi; od; # _Zerinvary Lajos_, Mar 19 2007
%p P := select(isprime,[$1..1609]): select(p->member(p-2,P),P); # _Peter Luschny_, Mar 03 2011
%p A006512 := proc(n)
%p 2+A001359(n) ;
%p end proc: # _R. J. Mathar_, Nov 26 2014
%t Select[Prime[Range[254]], PrimeQ[# - 2] &] (* _Robert G. Wilson v_, Jun 09 2005 *)
%t Transpose[Select[Partition[Prime[Range[300]], 2, 1], Last[#] - First[#] == 2 &]][[2]] (* _Harvey P. Dale_, Nov 02 2011 *)
%t Cases[Prime[Range[500]] + 2, _?PrimeQ] (* _Fred Patrick Doty_, Aug 23 2017 *)
%o (PARI) select(p->isprime(p-2),primes(1000))
%o (Magma) [n: n in PrimesUpTo(1610)|IsPrime(n-2)]; // _Bruno Berselli_, Feb 28 2011
%o (Haskell)
%o a006512 = (+ 2) . a001359 -- _Reinhard Zumkeller_, Feb 10 2015
%o (PARI) a(n)=p=3; while(p+2 < (p=nextprime(p+1)) || n-->0, ); p
%o vector(100, n, a(n)) \\ _Altug Alkan_, Dec 04 2015
%o (Python)
%o from sympy import primerange, isprime
%o print([n for n in primerange(1, 2001) if isprime(n - 2)]) # _Indranil Ghosh_, Jul 20 2017
%Y Subsequence of A139690.
%Y Bisection of A077800.
%Y Cf. A001097, A001359, A014574, A067829, A002476.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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