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%I
%S 1,2,3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36,39,41,44,48,50,
%T 51,53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,111,
%U 113,114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156
%N (Odd primes - 1)/2.
%C Or, numbers n such that 2n+1 is prime.
%C Also numbers not of the form 2xy+x+y. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
%C This sequence arises if you factor the product of a large number of the first odd numbers into the form 3^n(3)5^n(5)7^n(7)11^n(11)... Then n(3)/n(5) = 2, n(3)/n(7) = 3, n(3)/n(11) = 5,... . - Andrzej Staruszkiewicz (astar(AT)th.if.uj.edu.pl), May 31 2007
%C Kohen shows: A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d-1 chairs). We will show that there is a solution for every choice of the distances if and only if 2n+1 is a prime number [i.e. iff n is in A005097], using a theorem known as Combinatorial Nullstellensatz. [From _Jonathan Vos Post_, Jun 14 2010]
%C Starting from 6, positions at which new primes are seen for Goldbach partitions Eg 31 is first seen at 34 from 31+3, so position=1+(34-6)/2=15 [From _Bill McEachen_, Jul 05 2010]
%C Perfect error-correcting Lee codes of word length n over Z: it is conjectured that these always exist when 2n+1 is a prime, as mentioned in Horak [Jonathan Vos Post, Sep 19 2011].
%C Also solutions to: A000010(2*n+1) = n * A000005(2*n+1). - _Enrique Pérez Herrero_, Jun 07 2012
%C Solutions of the equation (2*n+1)'=1, where n' is the arithmetic derivative of n. [_Paolo P. Lava_, Nov 15 2012]
%C A193773(a(n)) = 1. - _Reinhard Zumkeller_, Jan 02 2013
%H T. D. Noe, <a href="/A005097/b005097.txt">Table of n, a(n) for n=1..1000</a>
%H Daniel Kohen and Ivan Sadofschi, <a href="http://arxiv.org/abs/1006.2571">A New Approach on the Seating Couples Problem</a>, Jun 13, 2010.
%H Dhananjay P. Mehendale, On Hamilton Decompositions, <a href="http://arxiv.org/abs/0806.0251">arXiv:0806.0251</a>
%H Peter Horak, Bader F. AlBdaiwi, <a href="http://arxiv.org/abs/1109.3475">Diameter Perfect Lee Codes</a>, arXiv:1109.3475v1 [cs.IT], Sep 15, 2011.
%F a(n) = A006093(n)/2 = A000010[A000040(n+1)]/2
%F a(n) = (p(n+1)^2-1)/(2*sigma(p(n+1))), for p(n)= n-th prime, =(A000040(n+1)^2-1)/(2*A000203(A000040(n+1))). - Gary Detlefs, May 02 2012
%F a(n) = (A065091(n) - 1) / 2. - _Reinhard Zumkeller_, Jan 02 2013
%p with(numtheory): p:=n-> ithprime(n):seq((p(n+1)^2-1)/(2*sigma(p(n+1))), n= 1..64) - Gary Detlefs, May 02 2012
%t Table[p=Prime[n];(p-1)/2, {n, 2, 22}] (* from _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)
%o (PARI) forprime(p=3,1e4,print1(p>>1", ")) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Haskell)
%o a005097 = (`div` 2) . a065091 -- _Reinhard Zumkeller_, Jan 02 2013
%Y Complement of A047845. Cf. A000040, A006005, A006093.
%Y A130290 is an essentially identical sequence.
%Y Numbers n such that 2n+k is prime: this seq(k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
%Y Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).
%K nonn,easy,changed
%O 1,2
%A _N. J. A. Sloane_.
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