cp's OEIS Frontend

A191374 Number of ways (up to rotations and reflections) of arranging numbers 1 through 2n around a circle such that the sum of each pair of adjacent numbers is composite.

%I A191374 #26 Apr 25 2016 12:05:02
%S A191374 0,0,1,44,912,61952,8160260,888954284,180955852060,50317255621843,
%T A191374 12251146829850324,4243527581615332664,1602629887788636447221,
%U A191374 622433536382831426225696,344515231090957672408413959
%N A191374 Number of ways (up to rotations and reflections) of arranging numbers 1 through 2n around a circle such that the sum of each pair of adjacent numbers is composite.
%C A191374 One of the obvious analogs of sequence A051252, which has the sums being prime. Presumably it is an open problem as to whether a(n) > 0 always for this problem as well.
%C A191374 The Guy reference deals with each adjacent pair summing to a prime. - _T. D. Noe_, Jun 08 2011
%D A191374 R. K. Guy, Unsolved Problems in Number Theory, section C1.
%F A191374 Bisection of A182540: a(n) = A182540(2*n). - _Max Alekseyev_, Aug 18 2013
%e A191374 a(3) = 1, the arrangement is 1,3,6,2,4,5.
%o A191374 (MATLAB)
%o A191374 function D=primecirc(n)
%o A191374 tic
%o A191374 a = 2:2*n;
%o A191374 A=perms(a);
%o A191374 for i =1:factorial(2*n-1)
%o A191374 B(i,:)=[1 A(i,:)];
%o A191374 end
%o A191374 for k=1:size(B,2)-1
%o A191374     F(:,k) = B(:,k)+B(:,k+1);
%o A191374 end
%o A191374 if k>1
%o A191374 F(:,k+1)=B(:,end)+B(:,1);
%o A191374 end
%o A191374 l=1;
%o A191374 for i=1:factorial(2*n-1)
%o A191374 if ~isprime(F(i,:)) == ones(1,length(B(1,:)))
%o A191374 C(l,:)=B(i,:);
%o A191374 l=l+1;
%o A191374 end
%o A191374 end
%o A191374 if ~exist('C')
%o A191374     D=0;
%o A191374     return
%o A191374 end
%o A191374 if size(C,1)==1
%o A191374 D=1;
%o A191374 else
%o A191374 D=size(C,1)/2;
%o A191374 end
%o A191374 toc
%Y A191374 Cf. A051252.
%K A191374 nonn,more
%O A191374 1,4
%A A191374 _Bennett Gardiner_, Jun 01 2011
%E A191374 a(8)-a(15) from _Max Alekseyev_, Aug 19 2013