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A051252
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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.
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14
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1, 1, 1, 2, 48, 512, 1440, 40512, 385072, 3154650, 106906168, 3197817022, 82924866213
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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
Is there a proof that this can always be done?
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 (noe(AT)sspectra.com), Jun 19 2002
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 (noe(AT)sspectra.com), Jul 01 2002
Prime circles can apparently be generated for any n using the Mathematica programs given in A072676 and A072184. - T. D. Noe (noe(AT)sspectra.com), Jul 08 2002
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 (pbotherstuff(AT)yahoo.co.uk), Oct 30 2007
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory, section C1.
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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EXAMPLE
| One arrangement for 2n=6 is 1,4,3,2,5,6 and this is essentially unique, so a(3)=1.
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MATHEMATICA
| $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
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CROSSREFS
| Cf. A000341, A070897, A072616, A072617, A072618, A072676.
Cf. A072184.
Sequence in context: A075690 A101362 A058090 * A005429 A035606 A157057
Adjacent sequences: A051249 A051250 A051251 * A051253 A051254 A051255
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KEYWORD
| nonn,nice
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AUTHOR
| Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu)
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