A242529 Number of cyclic arrangements (up to direction) of numbers 1,2,...,n such that any two neighbors are coprime.

1, 1, 1, 1, 6, 2, 36, 36, 360, 288, 11016, 3888, 238464, 200448, 3176496, 4257792, 402573312, 139511808, 18240768000, 11813990400, 440506183680, 532754620416, 96429560832000, 32681097216000, 5244692024217600, 6107246661427200, 490508471914905600, 468867166554931200, 134183696369843404800

Offset

1,5

Comments

a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S={1,2,...,n} of n elements and a specific pair-property P of "being coprime". For more details, see the link and A242519.

Links

Table of n, a(n) for n=1..29.

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

Wikipedia, Coprime integers

Formula

For n>2, a(n) = A086595(n)/2.

Example

There are 6 such cycles of length n=5: C_1={1,2,3,4,5}, C_2={1,2,3,5,4},
C_3={1,2,5,3,4}, C_4={1,2,5,4,3}, C_5={1,3,2,5,4}, and C_6={1,4,3,2,5}.
For length n=6, the count drops to just 2:
C_1={1,2,3,4,5,6}, C_2={1,4,3,2,5,6}.

Mathematica

A242529[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[Select[cpf[x], # != 1 &]];
cpf[x_] := Module[{i},
   Table[GCD[x[[i]], x[[i + 1]]], {i, Length[x] - 1}]];
Join[{1, 1}, Table[A242529[n], {n, 3, 10}]]
(* OR, a less simple, but more efficient implementation. *)
A242529[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[GCD[First[perm], Last[perm]] == 1, ct++];
     Return[ct],
     opt = remain; lr = Length[remain];
     For[i = 1, i <= lr, i++,
      new = First[opt]; opt = Rest[opt];
      If[GCD[Last[perm], new] != 1, Continue[]];
      A242529[n, Join[perm, {new}],
       Complement[Range[2, n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{1, 1}, Table[ct = 0; A242529[n, {1}, Range[2, n]]/2, {n, 3, 12}] ](* Robert Price, Oct 25 2018 *)

Prog

(C++) See the link.

Crossrefs

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242530, A242531, A242532, A242533, A242534.

Sequence in context: A201229 A038256 A373829 * A263088 A266231 A192355

Adjacent sequences: A242526 A242527 A242528 * A242530 A242531 A242532

Keyword

nonn,hard

Author

Stanislav Sykora, May 30 2014

Extensions

a(1) corrected, a(19)-a(29) added by Max Alekseyev, Jul 04 2014

Status

approved