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.

0, 0, 0, 0, 0, 2, 1, 4, 88, 0, 976, 22277, 22365, 376002, 3172018, 5821944, 10222624, 424452210, 6129894510, 38164752224

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..20.

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: A009830 A053374 A355517 * A093876 A375605 A322334

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

a(20) from Bert Dobbelaere, Feb 15 2020

Status

approved