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%I
%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 (hillcino368(AT)gmail.com), 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). [From _Farideh Firoozbakht_, Jul 04 2010]
%C Solutions of the equation (n-1)'+(n+1)'=2, where n' is the arithmetic derivative of n. [_Paolo P. Lava_, Dec 18 2012]
%C Subsequence of A217259 - numbers n such that antisigma(n+1) - antisigma(n-1) = 2*n - 1, where antisigma(m) = A024816(m) = sum of nondivisors of m. If n = average of twin prime pairs (q < p) then antisigma(p) - antisigma(q) = 2*n - 1 = p + q - 1. - _Jaroslav Krizek_, Mar 17 2013
%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="http://www.utm.edu/research/primes/lists/top20/twin.html">Twin Primes</a>
%H C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=TwinPrime">Twin primes</a>
%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 a(n) = A141515(k) iff A141515(k) -/+1 are both prime. [From _Giovanni Teofilatto_, Sep 19 2008]
%F A010051(a(n) - 1) * A010051(a(n) + 1) = 1. _Reinhard Zumkeller_, Apr 11 2012
%p P := select(isprime,[$1..1609]): map(p->p+1,select(p->member(p+2,P),P)); # _Peter Luschny_, Mar 03 2011
%p A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # _R. J. Mathar_, Jun 11 2011
%t Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* _Ray Chandler_, Oct 12 2005 *)
%o (PARI) p=2;forprime(q=3,1e4,if(q-p==2,print1(p+1", "));p=q) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Maxima) A014574(n) := block(
%o if n = 1 then
%o return(4),
%o p : A014574(n-1) ,
%o for k : 2 step 2 do (
%o if primep(p+k-1) and primep(p+k+1) then
%o return(p+k)
%o )
%o )$ /* _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
%Y Cf. A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.
%K nonn,easy,nice
%O 1,1
%A R. K. Guy, _N. J. A. Sloane_, _Eric W. Weisstein_
%E Offset changed to 1. - R. J. Mathar, Jun 11 2011
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