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%I #180 Feb 04 2024 01:10:22
%S 4,6,12,18,30,42,60,72,102,108,138,150,180,192,198,228,240,270,282,
%T 312,348,420,432,462,522,570,600,618,642,660,810,822,828,858,882,1020,
%U 1032,1050,1062,1092,1152,1230,1278,1290,1302,1320,1428,1452,1482,1488,1608
%N Average of twin prime pairs.
%C With an initial 1 added, this is the complement of the closure of {2} under a*b+1 and a*b-1. - _Franklin T. Adams-Watters_, Jan 11 2006
%C Also the square root of the product of twin prime pairs + 1. Two consecutive odd numbers can be written as 2k+1,2k+3. Then (2k+1)(2k+3)+1 = 4(k^2+2k+1) = 4(k+1)^2, a perfect square. Since twin prime pairs are two consecutive odd numbers, the statement is true for all twin prime pairs. - _Cino Hilliard_, May 03 2006
%C Or, single (or isolated) composites. Nonprimes k such that neither k-1 nor k+1 is nonprime. - _Juri-Stepan Gerasimov_, Aug 11 2009
%C Numbers n such that sigma(n-1) = phi(n+1). - _Farideh Firoozbakht_, Jul 04 2010
%C Aside from the first term in the sequence, all remaining terms have digital root 3, 6, or 9. - _J. W. Helkenberg_, Jul 24 2013
%C Numbers n such that n^2-1 is a semiprime. - _Thomas Ordowski_, Sep 24 2015
%C Every term but the first is a multiple of 6. - _Harvey P. Dale_, Mar 31 2023
%D Archimedeans Problems Drive, Eureka, 30 (1967).
%H T. D. Noe, <a href="/A014574/b014574.txt">Table of n, a(n) for n = 1..10000</a>
%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=TwinPrime">The Prime Glossary: Twin primes</a>
%H C. K. Caldwell, <a href="https://t5k.org/top20/page.php?id=1">The Top Twenty: Twin Primes</a>
%H Y. Fujiwara, <a href="http://dx.doi.org/10.1109/TIT.2013.2272695">Parsing a Sequence of Qubits</a>, IEEE Trans. Information Theory, 59 (2013), 6796-6806.
%H Y. Fujiwara, <a href="http://arxiv.org/abs/1207.1138">Parsing a Sequence of Qubits</a>, arXiv:1207.1138 [quant-ph], 2012-2013.
%H L. J. Gerstein, <a href="https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-reformulation-of-the-goldbach-conjecture">A reformulation of the Goldbach conjecture</a>, Math. Mag., 66 (1993), 44-45.
%H Brian Hayes, <a href="http://bit-player.org/2021/does-having-prime-neighbors-make-you-more-composite">Does having prime neighbors make you more composite?</a>, Bit-Player Article, Nov 04 2021
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%F a(n) = (A001359(n) + A006512(n))/2 = 2*A040040(n) = A054735(n)/2 = A111046(n)/4.
%F a(n) = A129297(n+4). - _Reinhard Zumkeller_, Apr 09 2007
%F A010051(a(n) - 1) * A010051(a(n) + 1) = 1. _Reinhard Zumkeller_, Apr 11 2012
%F a(n) = 6*A002822(n-1), n>=2. - _Ivan N. Ianakiev_, Aug 19 2013
%F a(n)^4 - 4*a(n)^2 = A062354(a(n)^2 - 1). - _Raphie Frank_, Oct 17 2013
%p P := select(isprime,[$1..1609]): map(p->p+1,select(p->member(p+2,P),P)); # _Peter Luschny_, Mar 03 2011
%p A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # _R. J. Mathar_, Jun 11 2011
%t Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* _Ray Chandler_, Oct 12 2005 *)
%t Mean/@Select[Partition[Prime[Range[300]],2,1],Last[#]-First[#]==2&] (* _Harvey P. Dale_, Jan 16 2014 *)
%o (PARI) p=2;forprime(q=3,1e4,if(q-p==2,print1(p+1", "));p=q) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Maxima) A014574(n) := block(
%o if n = 1 then
%o return(4),
%o p : A014574(n-1) ,
%o for k : 2 step 2 do (
%o if primep(p+k-1) and primep(p+k+1) then
%o return(p+k)
%o )
%o )$ /* _R. J. Mathar_, Mar 15 2012 */
%o (Haskell)
%o a014574 n = a014574_list !! (n-1)
%o a014574_list = [x | x <- [2,4..], a010051 (x-1) == 1, a010051 (x+1) == 1]
%o -- _Reinhard Zumkeller_, Apr 11 2012
%o (GAP) a:=1+Filtered([1..2000],p->IsPrime(p) and IsPrime(p+2)); # _Muniru A Asiru_, May 20 2018
%Y Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.
%Y A068507 is the intersection of A002182 and this sequence.
%K nonn,easy,nice
%O 1,1
%A _R. K. Guy_, _N. J. A. Sloane_, _Eric W. Weisstein_
%E Offset changed to 1 by _R. J. Mathar_, Jun 11 2011
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