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A227050 Number of essentially different ways of arranging numbers 1 through 2n around a circle so that the sum and absolute difference of each pair of adjacent numbers are prime. 5
0, 0, 0, 0, 0, 2, 1, 4, 88, 0, 976, 22277, 22365, 376002, 3172018, 5821944, 10222624, 424452210, 6129894510 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

See a similar problem, but for the set of numbers {0 through (n-1)}. - Stanislav Sykora, May 30 2014

LINKS

Table of n, a(n) for n=1..19.

Gary Antonick, Numberplay: Bernardo Recamán’s Primes in a Circle Puzzle, Jun 17 2013

Stanislav Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014; Table III.

EXAMPLE

For n = 6 the a(6) = 2 solutions are (1, 4, 9, 2, 5, 12, 7, 10, 3, 8, 11, 6) and (1, 6, 11, 8, 3, 10, 7, 4, 9, 2, 5, 12) because abs(1 - 4) = 3 and 1 + 4 = 5 are prime, etc.

MATHEMATICA

A227050[n_] :=

Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;

j1f[x_] := Join[{1}, x, {1}];

lpf[x_] := Length[

   Join[Select[asf[x], ! PrimeQ[#] &],

    Select[Differences[x], ! PrimeQ[#] &]]];

asf[x_] := Module[{i}, Table[x[[i]] + x[[i + 1]], {i, Length[x] - 1}]];

Table[A227050[n], {n, 1, 6}]

(* OR, a less simple, but more efficient implementation. *)

A227050[n_, perm_, remain_] := Module[{opt, lr, i, new},

   If[remain == {},

     If[PrimeQ[First[perm] - Last[perm]] &&

       PrimeQ[First[perm] + Last[perm]], ct++];

     Return[ct],

     opt = remain; lr = Length[remain];

     For[i = 1, i <= lr, i++,

      new = First[opt]; opt = Rest[opt];

      If[! (PrimeQ[Last[perm] - new] && PrimeQ[Last[perm] + new]),

       Continue[]];

      A227050[n, Join[perm, {new}],

       Complement[Range[2 n], perm, {new}]];

      ];

     Return[ct];

     ];

   ];

Table[ct = 0; A227050[n, {1}, Range[2, 2 n]]/2, {n, 1, 10}]

(* Robert Price, Oct 22 2018 *)

PROG

(C++) // Listed in the Sykora link.

CROSSREFS

Cf. similar sequences: A051252 (with sums of neighbors prime), A242527 (with sums of neighbors prime), A228626 (with differences of neighbors prime), A242528 (with sums and differences of neighbors prime).

Sequence in context: A061655 A009830 A053374 * A093876 A322334 A198371

Adjacent sequences:  A227047 A227048 A227049 * A227051 A227052 A227053

KEYWORD

nonn,more,hard

AUTHOR

Tim Cieplowski, Jun 29 2013

EXTENSIONS

a(15)-a(18) added by Tim Cieplowski, Jan 04 2015

a(19) from Fausto A. C. Cariboni, Jun 06 2017

STATUS

approved

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Last modified July 21 04:40 EDT 2019. Contains 325189 sequences. (Running on oeis4.)