A326494 Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.

1, 2, 4, 6, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset

0,2

Comments

The only allowed sets are the empty set, any singleton, any initial interval of positive integers and {2,4}. This can be shown by induction. - Andrew Howroyd, Aug 25 2019

Links

Table of n, a(n) for n=0..63.

Formula

a(n) = 2*n + 1 = A005408(n) for n > 3. - Andrew Howroyd, Aug 25 2019

Example

The a(0) = 1 through a(6) = 13 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {2}    {2}      {2}        {2}          {2}
           {1,2}  {3}      {3}        {3}          {3}
                  {1,2}    {4}        {4}          {4}
                  {1,2,3}  {1,2}      {5}          {5}
                           {2,4}      {1,2}        {6}
                           {1,2,3}    {2,4}        {1,2}
                           {1,2,3,4}  {1,2,3}      {2,4}
                                      {1,2,3,4}    {1,2,3}
                                      {1,2,3,4,5}  {1,2,3,4}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,5,6}

Mathematica

Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Union[Divide@@@Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&], Subtract@@@Select[Tuples[#, 2], Greater@@#&]]]&]], {n, 0, 10}]

Crossrefs

Subsets with difference are A054519.

Subsets with quotients are A326023.

Subsets with quotients > 1 are A326079.

Subsets without differences or quotients are A326490.

Cf. A005408, A007865, A051026, A325849, A326076, A326491.

Sequence in context: A024978 A191194 A191231 * A285593 A351631 A248898

Adjacent sequences: A326491 A326492 A326493 * A326495 A326496 A326497

Keyword

nonn

Author

Gus Wiseman, Jul 09 2019

Extensions

Terms a(20) and beyond from Andrew Howroyd, Aug 25 2019

Status

approved