%I A014574
%S A014574 4,6,12,18,30,42,60,72,102,108,138,150,180,192,198,228,240,270,282,312,
%T A014574 348,420,432,462,522,570,600,618,642,660,810,822,828,858,882,1020,1032,
%U A014574 1050,1062,1092,1152,1230,1278,1290,1302,1320,1428,1452,1482,1488,1608
%N A014574 Average of twin prime pairs.
%C A014574 With an initial 1 added, this is the complement of the closure of {2} 
               under a*b+1 and a*b-1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 11 2006
%C A014574 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 A014574 a(n) = A129297(n+2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 09 2007
%C A014574 Or, single (or isolated) composites. Also nonprimes k such that neither 
               k-1 nor k+1 is nonprime. - Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Aug 11 2009
%D A014574 Archimedeans Problems Drive, Eureka, 30 (1967).
%H A014574 T. D. Noe, <a href="b014574.txt">Table of n, a(n) for n=0..10000</a>
%H A014574 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/top20/
               twin.html">Twin Primes</a>
%H A014574 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=TwinPrime">
               Twin primes</a>
%H A014574 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TwinPrimes.html">Twin Primes</a>
%H A014574 Anonymous, <a href="http://www.mathematical.com/twindex2to1b.htm">Twin 
               Prime Pairs from 3 to 10000000(actually expressed as a(n)-+1)</a>
%H A014574 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica 
               de los numeros primos y perfectos</a>.
%F A014574 a(n) = {A001359(n+1) + A006512(n+1)}/2 = 2*A040040(n) = A054735(n+1)/
               2 = A111046(n+1)/4.
%F A014574 a(n) = A141515(n) iff A141515(n+1) -/+1 is both prime. [From Giovanni 
               Teofilatto (g.teofilatto(AT)tiscalinet.it), Sep 19 2008]
%p A014574 ZL:=[]:for p from 1 to 1610 do if (isprime(p) and isprime(p+2)) then 
               ZL:=[op(ZL),(((p+2)^2)-p^2)/4]; fi; od; print(ZL); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Mar 08 2007
%t A014574 Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] - Ray Chandler 
               (rayjchandler(AT)sbcglobal.net), Oct 12 2005
%Y A014574 Cf. A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.
%Y A014574 Sequence in context: A074998 A061715 A072570 this_sequence A034425 A073123 
               A079865
%Y A014574 Adjacent sequences: A014571 A014572 A014573 this_sequence A014575 A014576 
               A014577
%K A014574 nonn,easy,nice
%O A014574 0,1
%A A014574 R. K. Guy, N. J. A. Sloane (njas(AT)research.att.com), Eric Weisstein 
               (eric(AT)weisstein.com)