A242533 -id:A242533 - cp's OEIS frontend

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

Stanislav Sykora, May 30 2014

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.

			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}.

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

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

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 *)

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

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

A242530 Number of cyclic arrangements of S={1,2,...,2n} such that the binary expansions of any two neighbors differ by one bit.

Original entry on oeis.org

0, 0, 1, 0, 2, 8, 0, 0, 224, 754, 0, 26256, 0, 0, 22472304, 0, 90654576, 277251016, 0, 7852128780

Offset: 1

Stanislav Sykora, May 30 2014

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

In this case the property P is the Gray condition. The choice of the set S is important; when it is replaced by {0,1,2,...,2n-1}, the sequence changes completely and becomes A236602.

			The two cycles for n=5 (cycle length 10) are:
C_1={1,3,7,5,4,6,2,10,8,9}, C_2={1,5,4,6,7,3,2,10,8,9}.

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

Cf. A236602, A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242531, A242532, A242533, A242534.

A242530[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
btf[x_] := Module[{i},
   Table[DigitCount[BitXor[x[[i]], x[[i + 1]]], 2, 1], {i,
     Length[x] - 1}]];
lpf[x_] := Length[Select[btf[x], # != 1 &]];
Table[A242530[n], {n, 1, 5}]
 (* OR, a less simple, but more efficient implementation. *)
A242530[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[DigitCount[BitXor[First[perm], Last[perm]], 2, 1] == 1, ct++];
     Return[ct],
     opt = remain; lr = Length[remain];
     For[i = 1, i <= lr, i++,
      new = First[opt]; opt = Rest[opt];
      If[DigitCount[BitXor[Last[perm], new], 2, 1] != 1, Continue[]];
      A242530[n, Join[perm, {new}],
       Complement[Range[2, 2 n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Table[ct = 0; A242530[n, {1}, Range[2, 2 n]]/2, {n, 1, 10}] (* Robert Price, Oct 25 2018 *)

a(16)-a(20) from Fausto A. C. Cariboni, May 10 2017, May 15 2017

A242531 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is a divisor of their sum.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 3, 9, 26, 82, 46, 397, 283, 1675, 9938, 19503, 10247, 97978, 70478, 529383, 3171795, 7642285, 3824927, 48091810, 116017829, 448707198, 1709474581, 6445720883, 3009267707, 51831264296

Offset: 1

Stanislav Sykora, May 30 2014

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.

			The only such cycle of length n=5 is {1,2,4,5,3}.
For n=7 there are three solutions: C_1={1,2,4,5,7,6,3}, C_2={1,2,4,6,7,5,3}, C_3={1,2,6,7,5,4,3}.

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

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242532, A242533, A242534.

A242531[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
dvf[x_] := Module[{i},
   Table[Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
     Length[x] - 1}]];
lpf[x_] := Length[Select[dvf[x], ! # &]];
Join[{0, 1}, Table[A242531[n], {n, 3, 10}]]
(* OR, a less simple, but more efficient implementation. *)
A242531[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[Divisible[First[perm] + Last[perm],
       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[! Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
      A242531[n, Join[perm, {new}],
       Complement[Range[2, n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{0, 1}, Table[ct = 0; A242531[n, {1}, Range[2, n]]/2, {n, 3, 13}]] (* Robert Price, Oct 25 2018 *)

a(24)-a(28) from Fausto A. C. Cariboni, May 25 2017
a(29) from Fausto A. C. Cariboni, Jul 09 2020
a(30) from Fausto A. C. Cariboni, Jul 14 2020

A242532 Number of cyclic arrangements of S={2,3,...,n+1} such that the difference of any two neighbors is greater than 1, and a divisor of their sum.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 20, 39, 0, 0, 0, 0, 319, 967, 0, 0, 1464, 6114, 16856, 44370, 0, 0, 0, 0, 2032951, 8840796, 12791922, 101519154, 0, 0

Offset: 1

Stanislav Sykora, May 30 2014

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.
For this property P and sets {0,1,2,...,n-1} or {1,2,...,n} the problem does not appear to have any solution.
a(40)=a(41)=a(42)=a(43)=a(46)=a(47)=0. - Fausto A. C. Cariboni, May 17 2017

			The shortest such cycle is of length n=9: {2,4,8,10,5,7,9,3,6}.
The next a(n)>0 occurs for n=14 and has 20 solutions.
The first and the last of these are:
C_1={2,4,8,10,5,7,14,12,15,13,11,9,3,6},
C_2={2,4,12,15,13,11,9,3,5,7,14,10,8,6}.

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

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242533, A242534.

A242532[n_] := Count[Map[lpf, Map[j2f, Permutations[Range[3, n + 1]]]], 0]/2;
j2f[x_] := Join[{2}, x, {2}];
dvf[x_] := Module[{i},
   Table[Abs[x[[i]] - x[[i + 1]]] > 1 &&
     Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
     Length[x] - 1}]];
lpf[x_] := Length[Select[dvf[x], ! # &]];
Table[A242532[n], {n, 1, 10}]
(* OR, a less simple, but more efficient implementation. *)
A242532[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[Abs[First[perm] - Last[perm]] > 1 &&
       Divisible[First[perm] + Last[perm], 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[Abs[Last[perm] - new] <= 1 || !
         Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
      A242532[n, Join[perm, {new}],
       Complement[Range[3, n + 1], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Table[ct = 0; A242532[n, {2}, Range[3, n + 1]]/2, {n, 1, 15}] (* Robert Price, Oct 25 2018 *)

a(29)-a(37) from Fausto A. C. Cariboni, May 17 2017

A242534 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is not coprime to their sum.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 72, 288, 3600, 17856, 174528, 2540160, 14768640, 101030400, 1458266112, 11316188160, 140951577600, 2659218508800, 30255151463424, 287496736542720, 5064092578713600, 76356431941939200, 987682437203558400, 19323690313219522560

Offset: 1

Stanislav Sykora, May 30 2014

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.

Compare this with A242533 where the property is inverted.

			The first and the last of the 72 cycles for n=10 are:
C_1={1,3,5,10,2,4,8,6,9,7} and C_72={1,7,5,10,8,4,2,6,3,9}.
There are no solutions for cycle lengths from 2 to 9.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..27
S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533.

A242534[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[Select[cpf[x], ! # &]];
cpf[x_] := Module[{i},
   Table[! CoprimeQ[x[[i]] - x[[i + 1]], x[[i]] + x[[i + 1]]], {i,
     Length[x] - 1}]];
Join[{1}, Table[A242534[n], {n, 2, 10}]]
(* OR, a less simple, but more efficient implementation. *)
A242534[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[!
       CoprimeQ[First[perm] + Last[perm], 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[CoprimeQ[Last[perm] + new, Last[perm] - new], Continue[]];
      A242534[n, Join[perm, {new}],
       Complement[Range[2, n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{1}, Table[ct = 0; A242534[n, {1}, Range[2, n]]/2, {n, 2, 12}] ](* Robert Price, Oct 25 2018 *)

a(19)-a(27) from Hiroaki Yamanouchi, Aug 30 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A242530 Number of cyclic arrangements of S={1,2,...,2n} such that the binary expansions of any two neighbors differ by one bit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A242531 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is a divisor of their sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A242532 Number of cyclic arrangements of S={2,3,...,n+1} such that the difference of any two neighbors is greater than 1, and a divisor of their sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A242534 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is not coprime to their sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions