A326494 -id:A326494 - cp's OEIS frontend

A326491 Number of maximal subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

1, 1, 2, 2, 3, 4, 5, 7, 9, 10, 16, 22, 27, 39, 52, 70, 90, 120, 150, 198, 262, 357, 448, 602, 782, 1004, 1294, 1715, 2229, 2932, 3698, 4844, 6259, 8188, 10274, 13446, 16895, 21954, 27470, 35843, 45411, 58949, 73940, 95200, 120594, 154511, 192996, 247967, 312643

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(1) = 1 through a(9) = 10 subsets:
  {1}  {1}  {1}    {1}    {1}      {1}      {1}        {1}        {1}
       {2}  {2,3}  {2,3}  {2,3}    {2,3}    {2,3,7}    {2,5,6}    {2,6,7}
                   {3,4}  {2,5}    {2,5,6}  {2,5,6}    {2,5,8}    {3,4,5}
                          {3,4,5}  {3,4,5}  {2,6,7}    {2,6,7}    {3,5,7}
                                   {4,5,6}  {3,4,5}    {3,4,5}    {2,3,7,8}
                                            {3,5,7}    {3,5,7}    {2,5,6,9}
                                            {4,5,6,7}  {2,3,7,8}  {2,5,8,9}
                                                       {4,5,6,7}  {4,5,6,7}
                                                       {5,6,7,8}  {4,6,7,9}
                                                                  {5,6,7,8,9}

Subsets without differences or quotients are A326490.
Subsets with differences and quotients are A326494.
Maximal subsets without differences are A121269
Maximal subsets without quotients are A326492.
Cf. A007865, A051026, A054519, A326023, A326117, A326489, A326496, A327591.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Union[Divide@@@Reverse/@Subsets[#,{2}],Subtract@@@Reverse/@Subsets[#,{2}]]]=={}&]]],{n,0,10}]

a(n) = A326497(n) + 1 for n > 1. - Andrew Howroyd, Aug 30 2019

a(16)-a(40) from Andrew Howroyd, Aug 30 2019

a(41)-a(48) from Jinyuan Wang, Mar 04 2025

A326490 Number of subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 18, 31, 46, 72, 102, 172, 259, 428, 607, 989, 1329, 2142, 3117, 4953, 6956, 11032, 15321, 23979, 33380, 48699, 66849, 104853, 144712, 220758, 304133, 461580, 636556, 973843, 1316513, 1958828, 2585433, 3882843, 5237093, 7884277, 10555739, 15729293

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {2,3}  {4}    {4}      {4}
                       {2,3}  {5}      {5}
                       {3,4}  {2,3}    {6}
                              {2,5}    {2,3}
                              {3,4}    {2,5}
                              {3,5}    {2,6}
                              {4,5}    {3,4}
                              {3,4,5}  {3,5}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,5,6}
                                       {3,4,5}
                                       {4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..90

Subsets without difference are A007865.
Maximal subsets without differences or quotients are A326491.
Subsets without quotients are A327591.
Subsets with differences and quotients are A326494.
Cf. A051026, A054519, A325860, A326023, A326079, A326489.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Union[Divide@@@Reverse/@Subsets[#,{2}],Subtract@@@Reverse/@Subsets[#,{2}]]]=={}&]],{n,0,10}]

a(n)={
   my(recurse(k, b)=
    if(k > n, 1,
      my(t = self()(k + 1, b));
      for(i=1, k\2, if(bittest(b,i) && (bittest(b,k-i) || (!(k%i) && bittest(b,k/i))), return(t)));
      t += self()(k + 1, b + (1<Andrew Howroyd, Aug 25 2019

For n > 0, a(n) = A326495(n) + 1.

a(19)-a(41) from Andrew Howroyd, Aug 25 2019

cp's OEIS Frontend

A326491 Number of maximal subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

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