cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, May 30 2014

Keywords

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.

Examples

			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}.

Links

Crossrefs

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

Programs

  • 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 *)

Formula

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

Extensions

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

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

Views

Author

Stanislav Sykora, May 30 2014

Keywords

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.

Examples

			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}.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Stanislav Sykora, May 30 2014

Keywords

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.
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

Examples

			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}.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

A242533 Number of cyclic arrangements of S={1,2,...,2n} such that the difference of any two neighbors is coprime to their sum.

Original entry on oeis.org

1, 1, 2, 36, 288, 3888, 200448, 4257792, 139511808, 11813990400, 532754620416
Offset: 1

Views

Author

Stanislav Sykora, May 30 2014

Keywords

Comments

a(n)=NPC(2n;S;P) is the count of all neighbor-property cycles for a specific set S of 2n elements and a specific pair-property P. For more details, see the link and A242519.
Conjecture: in this case it seems that NPC(n;S;P)=0 for all odd n, so only the even ones are listed. This is definitely not the case when the property P is replaced by its negation (see A242534).

Examples

			For n=4, the only cycle is {1,2,3,4}.
The two solutions for n=6 are: C_1={1,2,3,4,5,6} and C_2={1,4,3,2,5,6}.

Links

Crossrefs

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242534.

Programs

  • Mathematica
    A242533[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 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[A242533[n], {n, 2, 5}]]
(* OR, a less simple, but more efficient implementation. *)
A242533[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[]];
      A242533[n, Join[perm, {new}],
       Complement[Range[2, 2 n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{1}, Table[ct = 0; A242533[n, {1}, Range[2, 2 n]]/2, {n, 2, 6}] ](* Robert Price, Oct 25 2018 *)

Extensions

a(10)-a(11) from Fausto A. C. Cariboni, May 31 2017, Jun 01 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

Views

Author

Stanislav Sykora, May 30 2014

Keywords

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.
Compare this with A242533 where the property is inverted.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

a(19)-a(27) from Hiroaki Yamanouchi, Aug 30 2014
Previous Showing 11-15 of 15 results.

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