A242528 Number of cyclic arrangements of {0,1,...,n-1} such that both the difference and the sum of any two neighbors are prime.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 18, 13, 62, 8, 133, 225, 209, 32, 2644, 4462, 61341, 113986, 750294, 176301, 7575912, 3575686, 7705362, 36777080, 108638048, 97295807

Offset: 1

Author

Stanislav Sykora, May 30 2014

Keywords

• nonn
• hard
• more

Comments

a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519.

In this case the set is S={0 through n-1}. For the same pair-property P but the set S={1 through n}, see A227050.

Examples

			For n=12 (the first n for which a(n)>0) there are two such cycles:
C_1={0, 5, 2, 9, 4, 1, 6, 11, 8, 3, 10, 7},
C_2={0, 7, 10, 3, 8, 5, 2, 9, 4, 1, 6, 11}.
		

Links

• S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.

Crossrefs

Cf. A227050, A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242529, A242530, A242531, A242532, A242533, A242534.

Programs

Mathematica

A242528[n_] :=
Count[Map[lpf, Map[j0f, Permutations[Range[n - 1]]]], 0]/2;
j0f[x_] := Join[{0}, x, {0}];
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[A242528[n], {n, 1, 8}]
(* OR, a less simple, but more efficient implementation. *)
A242528[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[]];
      A242528[n, Join[perm, {new}],
       Complement[Range[n - 1], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Table[ct = 0; A242528[n, {0}, Range[n - 1]]/2, {n, 1, 18}]
(* Robert Price, Oct 22 2018 *)

Extensions

a(29)-a(33) from Fausto A. C. Cariboni, May 20 2017