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.

0, 0, 1, 44, 912, 61952, 8160260, 888954284, 180955852060, 50317255621843, 12251146829850324, 4243527581615332664, 1602629887788636447221, 622433536382831426225696, 344515231090957672408413959

Offset: 1

Author

Bennett Gardiner, Jun 01 2011

Keywords

• nonn
• more

Comments

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.

The Guy reference deals with each adjacent pair summing to a prime. - T. D. Noe, Jun 08 2011

Examples

			a(3) = 1, the arrangement is 1,3,6,2,4,5.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, section C1.

Crossrefs

Cf. A051252.

Programs

MATLAB

function D=primecirc(n)
tic
a = 2:2*n;
A=perms(a);
for i =1:factorial(2*n-1)
B(i,:)=[1 A(i,:)];
end
for k=1:size(B,2)-1
    F(:,k) = B(:,k)+B(:,k+1);
end
if k>1
F(:,k+1)=B(:,end)+B(:,1);
end
l=1;
for i=1:factorial(2*n-1)
if ~isprime(F(i,:)) == ones(1,length(B(1,:)))
C(l,:)=B(i,:);
l=l+1;
end
end
if ~exist('C')
    D=0;
    return
end
if size(C,1)==1
D=1;
else
D=size(C,1)/2;
end
toc

Formula

Bisection of A182540: a(n) = A182540(2*n). - Max Alekseyev, Aug 18 2013

Extensions

a(8)-a(15) from Max Alekseyev, Aug 19 2013