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%I #49 Jun 07 2016 07:05:53
%S 1,1,1,2,48,512,1440,40512,385072,3154650,106906168,3197817022,
%T 82924866213,4025168862425,127854811616691
%N Number of essentially different ways of arranging numbers 1 through 2n around a circle so that sum of each pair of adjacent numbers is prime.
%C _Jud McCranie_ reports that he was able to find a solution for each n <= 225 (2n <= 450) in just a few seconds. - Jul 05 2002
%C Is there a proof that this can always be done?
%C The Mathematica program for this sequence uses backtracking to find all solutions for a given n. To verify that at least one solution exists for a given n, the backtracking function be made to stop when the first solution is found. Solutions have been found for n <= 48. - _T. D. Noe_, Jun 19 2002
%C This sequence is from the prime circle problem. There is no known proof that a(n) > 0 for all n. However, for many n (see A072618 and A072676), we can prove that a(n) > 0. Also, the sequence A072616 seems to imply that there are always solutions in which the odd (or even) numbers are in order around the circle. - _T. D. Noe_, Jul 01 2002
%C Prime circles can apparently be generated for any n using the Mathematica programs given in A072676 and A072184. - _T. D. Noe_, Jul 08 2002
%C The following seems to always produce a solution: Work around the circle starting with 1 but after that always choosing the largest remaining number that fits. For example, if n = 4 this gives 1, 6, 7, 4, 3, 8, 5, 2. See A088643 for a sequence on a related idea. - _Paul Boddington_, Oct 30 2007
%C See A228917 for a similar conjecture on twin primes. - _Zhi-Wei Sun_, Sep 08 2013
%C See A242527 for a similar problem on the set of numbers {0 through (n-1)}. - _Stanislav Sykora_, May 30 2014
%C James Tilley and Stan Wagon report that all terms up to n = 10^6 are nonzero. _Charles R Greathouse IV_, Feb 05 2016
%D R. K. Guy, Unsolved Problems in Number Theory, second edition, Springer, 1994. See section C1.
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On Neighbor-Property Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCircle.html">Prime Circle.</a>
%e One arrangement for 2n=6 is 1,4,3,2,5,6 and this is essentially unique, so a(3)=1.
%t $RecursionLimit=500; try[lev_] := Module[{t, j}, If[lev>2n, (*then make sure the sum of the first and last is prime*) If[PrimeQ[soln[[1]]+soln[[2n]]]&&soln[[2]]<=soln[[2n]], (*Print[soln]; *) cnt++ ], (*else append another number to the soln list*) t=soln[[lev-1]]; For[j=1, j<=Length[s[[t]]], j++, If[ !MemberQ[soln, s[[t]][[j]]], soln[[lev]]=s[[t]][[j]]; try[lev+1]; soln[[lev]]=0]]]]; For[lst={}; n=1, n<=7, n++, s=Table[{}, {2n}]; For[i=1, i<=2n, i++, For[j=1, j<=2n, j++, If[i!=j&&PrimeQ[i+j], AppendTo[s[[i]], j]]]]; soln=Table[0, {2n}]; soln[[1]]=1; cnt=0; try[2]; AppendTo[lst, cnt]]; lst (* _T. D. Noe_ *)
%o (C++) listed in the link (S. Sykora)
%Y Cf. A000341, A070897, A072616, A072617, A072618, A072676, A072184, A103839, A227050, A228917, A242527, A242528.
%K nonn,nice
%O 1,4
%A _Jud McCranie_
%E a(14)-a(15) from _Max Alekseyev_, Sep 19 2013
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