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 A014574 Average of twin prime pairs. 325
 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 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 Or, single (or isolated) composites. Nonprimes k such that neither k-1 nor k+1 is nonprime. - Juri-Stepan Gerasimov, Aug 11 2009 Numbers n such that sigma(n-1) = phi(n+1). - Farideh Firoozbakht, Jul 04 2010 Solutions of the equation (n-1)'+(n+1)'=2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012 Aside from the first term in the sequence, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013 Numbers n such that n^2-1 is a semiprime. - Thomas Ordowski, Sep 24 2015 REFERENCES Archimedeans Problems Drive, Eureka, 30 (1967). LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 C. K. Caldwell, The Prime Glossary: Twin primes C. K. Caldwell, The Top Twenty: Twin Primes Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806. Y. Fujiwara, Parsing a Sequence of Qubits, arXiv:1207.1138 [quant-ph], 2012-2013. L. J. Gerstein, A reformulation of the Goldbach conjecture, Math. Mag., 66 (1993), 44-45. Eric Weisstein's World of Mathematics, Twin Primes FORMULA a(n) = (A001359(n) + A006512(n))/2 = 2*A040040(n) = A054735(n)/2 = A111046(n)/4. a(n) = A129297(n+4). - Reinhard Zumkeller, Apr 09 2007 A010051(a(n) - 1) * A010051(a(n) + 1) = 1. Reinhard Zumkeller, Apr 11 2012 a(n) = 6*A002822(n-1), n>=2. - Ivan N. Ianakiev, Aug 19 2013 a(n)^4 - 4*a(n)^2 = A062354(a(n)^2 - 1). - Raphie Frank, Oct 17 2013 MAPLE P := select(isprime, [\$1..1609]): map(p->p+1, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011 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 MATHEMATICA Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* Ray Chandler, Oct 12 2005 *) Mean/@Select[Partition[Prime[Range], 2, 1], Last[#]-First[#]==2&] (* Harvey P. Dale, Jan 16 2014 *) PROG (PARI) p=2; forprime(q=3, 1e4, if(q-p==2, print1(p+1", ")); p=q) \\ Charles R Greathouse IV, Jun 10 2011 (Maxima) A014574(n) := block(     if n = 1 then         return(4),     p : A014574(n-1) ,     for k : 2 step 2 do (         if primep(p+k-1) and primep(p+k+1) then             return(p+k)     ) )\$ /* R. J. Mathar, Mar 15 2012 */ (Haskell) a014574 n = a014574_list !! (n-1) a014574_list = [x | x <- [2, 4..], a010051 (x-1) == 1, a010051 (x+1) == 1] -- Reinhard Zumkeller, Apr 11 2012 (GAP) a:=1+Filtered([1..2000], p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018 CROSSREFS Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046. Sequence in context: A280469 A072570 A217259 * A258838 A034425 A073123 Adjacent sequences:  A014571 A014572 A014573 * A014575 A014576 A014577 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Offset changed to 1 by R. J. Mathar, Jun 11 2011 STATUS approved

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Last modified October 21 23:47 EDT 2020. Contains 337948 sequences. (Running on oeis4.)